Home Mathematics Hamilton cycles in almost distance-hereditary graphs
Article Open Access

Hamilton cycles in almost distance-hereditary graphs

  • Bing Chen and Bo Ning EMAIL logo
Published/Copyright: February 9, 2016
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 14 Issue 1

Abstract

Let G be a graph on n ≥ 3 vertices. A graph G is almost distance-hereditary if each connected induced subgraph H of G has the property dH(x, y) ≤ dG(x, y) + 1 for any pair of vertices x, y ∈ V(H). Adopting the terminology introduced by Broersma et al. and Čada, a graph G is called 1-heavy if at least one of the end vertices of each induced subgraph of G isomorphic to K1,3 (a claw) has degree at least n/2, and is called claw-heavy if each claw of G has a pair of end vertices with degree sum at least n. In this paper we prove the following two theorems: (1) Every 2-connected, claw-heavy and almost distance-hereditary graph is Hamiltonian. (2) Every 3-connected, 1-heavy and almost distance-hereditary graph is Hamiltonian. The first result improves a previous theorem of Feng and Guo [J.-F. Feng and Y.-B. Guo, Hamiltonian cycle in almost distance-hereditary graphs with degree condition restricted to claws, Optimazation57 (2008), no. 1, 135–141]. For the second result, its connectedness condition is sharp since Feng and Guo constructed a 2-connected 1-heavy graph which is almost distance-hereditary but not Hamiltonian.

Keywords: Hamilton cycle; Almost distance-hereditary graph; Claw-free graph; 1-heavy graph; 2-heavy graph; Claw-heavy graph
MSC 2010: 05C38; 05C45

1 Introduction

In this paper, we only consider the graphs which are finite, undirected and without multi-edges and loops. For terminology and notation not defined here, we refer to Bondy and Murty [1].

Let G be a graph with vertex set V(G) and H be a subgraph of G. For a vertex υ ∈ V(G), we denote by NH(υ) the set of vertices which are adjacent to υ in H, and by dH(υ) = |NH(υ)| the degree of υ in H. For two vertices x, y ∈ V(G), an (x, y)-path in H is a path starting from x to y with all vertices in H. The distance of x and y in H, denoted by dH(x, y), is defined as the length of a shortest (x, y)-path in H. When there is no danger of ambiguity, we use N(υ), d(υ) and d(x, y) instead of NG(υ), dG(υ) and dG(x, y), respectively.

A graph is called Hamiltonian if it contains a Hamilton cycle, i.e., a cycle passing through all the vertices of the graph. The study of cycles, especially Hamilton cycles, may be one of the most important and most studied areas of graph theory. It is well-known that to determine whether a given graph contains a Hamilton cycle is NP-complete, which is shown by Karp [2]. However, if we only consider some restricted graph classes, then the situation is completely changed. A graph G is called distance-hereditary if each connected induced subgraph H has the property that dH(x, y) = dG(x, y) for any pair of vertices x, y in H. This concept was introduced by Howorka [3] and a complete characterization of distance-hereditary graphs could be found in [3]. In 2002, Hsieh, Ho, Hsu and Ko [4] obtained an O(|V| + |E|)-time algorithm to solve the Hamiltonian problem on distance-hereditary graphs. Some other optimization problems can also be solved in linear time for distance-hereditary graphs although they are proved to be NP-hard for general graphs. For references in this direction, we refer to [5, 6].

A graph G is called almost distance-hereditary if each connected induced subgraph H of G has the property dH(x, y) ≤ dG(x, y) + 1 for any pair of vertices x, y ∈ V(H). For some properties and a characterization of almost-distance hereditary graphs, we refer to [7].

Let G be a graph. An induced subgraph of G isomorphic to K1,3 is called a claw. The vertex of degree 3 in the claw is called its center and the other vertices are its end vertices. G is called claw-free if G contains no claw. Throughout this paper, whenever the vertices of a claw are listed, its center is always the first one.

Many results about the existence of Hamilton cycles in claw-free graphs have been obtained. In particular, Feng and Guo [8] gave the following result on Hamiltonicity of almost distance-hereditary claw-free graphs.

Theorem 1.1

(Feng and Guo [8]). Let G be a 2-connected claw-free graph. If G is almost distance-hereditary, then G is Hamiltonian.

Let G be a graph on n vertices. A vertex υ of G is called heavy if d(υ) ≥ n/2. Broersma et al. [9] introduced the concepts of 1-heavy graph and 2-heavy graph. Later, Fujisawa and Yamashita [10] and Čada [11] introduced the concept of claw-heavy graphs, independently. Following [9–11], we say that a claw in G is 1-heavy (2-heavy) if at least one (two) of its end vertices is (are) heavy. G is called 1-heavy (2-heavy) if every claw of it is 1-heavy (2-heavy), and called claw-heavy if every claw of it has two end vertices with degree sum at least n. It is easily seen that every claw-free graph is 1-heavy (2-heavy, claw-heavy), every 2-heavy graph is claw-heavy and every claw-heavy graph is 1-heavy. But not every claw-heavy graph is 2-heavy, and not every 1-heavy graph is claw-heavy.

In [12], Feng and Guo extended Theorem 1.1 to a larger graph class of 2-heavy graphs.

Theorem 1.2

(Feng and Guo [12]). Let G be a 2-connected 2-heavy graph. If G is almost distance-hereditary, then G is Hamiltonian.

Feng and Guo [12] also constructed a 2-connected 1-heavy graph which is almost distance-hereditary but not Hamiltonian. Thus it is natural to ask which is the minimum connectivity for a 1-heavy almost distance-hereditary graph under this connectivity condition to be Hamiltonian.

Motivated by [9, 13–15], in this paper we obtain the following two theorems which extend Theorem 1.1 and Theorem 1.2. In particular, Theorem 1.3 improves Theorem 1.2, and Theorem 1.4 answers the problem proposed above.

Theorem 1.3.

Let G be a 2-connected claw-heavy graph. If G is almost distance-hereditary, then G is Hamiltonian.

Theorem 1.4.

Let G be a 3-connected 1-heavy graph. If G is almost distance-hereditary, then G is Hamiltonian.

We emphasize that our technique of proofs is different from Feng and Guo [12]. One of our main tools is the so called "Ore-cycle" (motivated by Lemma 3 in [13]) introduced by Li et al. [16].

Remark 1.5.

The graph in Fig. 1 shows that the result in Theorem 1.3 indeed strengthen that in Theorem 1.2. As shown in [13, Fig. 2], let n ≥ 10 be an even integer and Kn/2 + Kn/2−3denote the join of two complete graphs Kn/2and Kn/2−3. Choose a vertex y ∈ V(Kn/2) and construct a graph G with V(G) = V(Kn/2 + Kn/2−3) ∪ {υ, u, x} and E(G) = E(Kn/2 + Kn/2−3) ∪ {uυ, uy, ux} ∪ {υw, xw : w ∈ V(Kn/2−3)}. It is easy to see that G is a Hamiltonian graph satisfying the condition of Theorem 1.3, but not the condition of Theorem 1.2.

We postpone the proofs of Theorem 1.3 and 1.4 to Section 3.

Fig. 1. A graph which shows the result in Theorem 1.3 strengthens that in Theorem 1.2.
Fig. 1.

A graph which shows the result in Theorem 1.3 strengthens that in Theorem 1.2.

2 Preliminaries

Let G be a graph on n vertices and k ≥ 3 be an integer. Recall that a vertex of degree at least n/2 in G is a heavy vertex; otherwise it is light. A claw in G is called a light claw if all its end vertices are light, and is called an o-light claw if any pair of end vertices has degree sum less than n. A cycle in G is called a heavy cycle if it contains all heavy vertices of G. Following [16], we use Ẽ(G) to denote the set {uυ : uυ ∈ E(G) or d(u) + d(υ) ≥ n, u, υ ∈ V(G)}. A sequence of vertices C = υ1υ2 … υk υ1 is called an Ore-cycle or briefly, o-cycle of G, if we have υiυi+1 ∈ Ẽ(G) for every i ∈ {1, 2, · , k}, where υ1 = υk+1.

Let G be a graph and k be a nonnegative integer. For a cycle C of G and a vertex u ∈ V(G)\V(C), a subgraph F of G is said to be a (u, C ; k)-fanif F is a union of paths P1, P2, … , Pk, where Pi is a (u, wi)-path (1 ≤ i ≤ k), Pi ∩ C = {wi} (1 ≤ i ≤ k) and Pi ∩ Pj = {u} for 1 ≤ i < j ≤ k. In the following, we use F = (u: P1, P2, … , Pk) to denote the fan. The vertices in V(F)\{w1, w2, … , wk} are called internal vertices of F.

We need some notations from [13]. Let H be a path or a cycle with a given orientation. We denote by H← the same graph as H but with the reverse orientation. For a vertex υ ∈ V(H), we use υH+ to denote the successor of υ on H, and υH− to denote its predecessor. If S ⊆ V(H), then define S+H = {υ+H : υ ∈ S} and S−H = {υ−H : υ ∈ S}. If there is no danger of ambiguity, we denote υ+H, υ−H, S+H and S−H by υ+, υ−, S+ and S−, respectively. For two vertices u, υ ∈ V(H), we denote by H[u, υ] the segment of H from u to υ, and denote H(u, υ), H [u, υ) and H(u, υ] by the paths H[u, υ] − {u, υ}, H[u, υ] − {υ} and H[u, υ] − {u}, respectively.

To prove Theorems 1.3 and 1.4, the following six lemmas are needed. In particular, similar proofs of the facts in Lemma 2.3 can be found in [13, 16] (for example, see Claims 1-4 of Theorem 8 in [16]). For the convenience of the readers, we write the detailed proofs here.

Lemma 2.1

(Bollobás and Brightwell [17], Shi [18]). Every 2-connected graph contains a heavy cycle.

Lemma 2.2

(Li, Wang, Ryjáček, Zhang [16]). Let G be graph and let C′ be an o-cycle of G. Then there exists a cycle C of G such that V(C′) ⊆ V(C).

Lemma 2.3.

Let G be a non-Hamiltonian graph on n vertices, C be a longest cycle (a longest heavy cycle) of G, R a component of G − V(C), and A = {υ1, υ2, … , υ3} the set of neighbors of R on C. Let u ∈ V(R) andυi, υj ∈ A. Then there hold

(a) uυ−i ∉ Ẽ(G), uυ+i ∉ Ẽ(G); (b) υ−iυ−j ∉ Ẽ(G), υ+iυ+j ∉ Ẽ(G); and (c)ifυ−iυ+i ∉ Ẽ(G), thenυiυ−j ∉ Ẽ(G), υiυ+j ∉ Ẽ(G).

Furthermore, if G is a 2-connected claw-heavy graph, then

(d) υ−iυ+i ∈ Ẽ(G) andυ−jυ+j ∈ Ẽ(G); (e) υ−iυ+i ∈ E(G) orυ−jυ+j ∈ E(G).

Proof. (a) Suppose that uυ−i ∈ Ẽ(G). Then C′ = uυiC[υi, υ−i]υ−iu is an o-cycle of length longer than C, a contradiction. The other assertion can be proved similarly.

(b) Suppose that υ−iυ−j ∈ Ẽ(G). Then C′=uυjC[υj,υi−]υi−υj−C←[υj−,υi]υiu is an o-cycle of length longer than C, a contradiction. The other assertion can be proved similarly.

(c) Suppose that υiυ−j ∈ Ẽ(G). Then C′ = υiuυjC[υj, υ−i]υ−iυ+iC[υ+i, υ−i]υ−iυi is an o-cycle of length longer than C, a contradiction. The other assertion can be proved similarly.

(d) If υ−iυ+i ∉ E(G), then by (a), we know uυ−i ∉ E(G), uυ+i ∉ E(G). Thus {υi, u, υi−, υi+}, induces a claw. Since G is claw-heavy, by (a), we have d(υ−i) + d(υ+i) ≥ n. This implies that υ−iυ+i ∈ Ẽ(G). If υ−iυ+i ∈ E(G), then obviously υ−iυ+i ∈ Ẽ(G). The other assertion can be proved similarly.

(e) Suppose that υ−iυ+i ∉ E(G) and υ−jv+j ∉ E(G). By (d), we have d(υ−i) + d(υ+i) ≥ n and d(υ−j) + d(υ+j) ≥ n. This implies that d(υ−i) + d(υ−j) ≥ n or d(υ+i) + d(υ+j) ≥ n. Thus υ−iυ−j ∈ Ẽ(G) or υ+iυ+j ∈ Ẽ(G), a contradiction to (b).

Lemma 2.4.

Let G be a non-Hamiltonian graph, C be a longest cycle (a longest heavy cycle) of G, R a component of G − V(C), and A = {υ1, υ2, . . . , υk} the set of neighbors of R on C. Let υi, υj ∈ A. Then there hold

(a) forl ∈ V(C(υi, υ−j]), ifυ−il ∈ Ẽ(G), thenl−υ+j ∉ Ẽ(G) andl+υ+j ∉ Ẽ(G);

(b) forl ∈ V(C(υi, υ−j]) ∩ N(υi), ifυ−iυ+i ∈ Ẽ(G), thenl−υ−j ∉ Ẽ(G) andl+υ+j ∉ Ẽ(G); and

(c) forl ∈ V(C(υi, υ−j]) ∩ N(υi) ∩ N(υ−j), ifυ−iυ+i ∈ Ẽ(G), thenl−l+ ∉ Ẽ(G).

Proof. Let P be a (υi, υj)-path with all internal vertices in R.

(a) Suppose l−υ+j ∈ Ẽ(G). Then C' = P←C[υi, l−]l−υj+C[υj+, υi−]υi−lC[l, υj] is an o-cycle such that V(C) ⊂ V(C′). By Lemma 2.2, there is a longer cycle C″ containing all vertices in C, that is, a longer cycle (a longer heavy cycle) in G, contradicting the choice of C. Suppose l+υ+j ∈ Ẽ(G). Then C' = PC←[υj, l+]l+υj+C[υj+, υi−]υi−lC←[l, υi] is an o-cycle such that V(C) ⊂ V(C′), a contradiction.

(b) Suppose l−υ−j ∈ Ẽ(G). Then C' = PC[υj, υi−]υi−υi+C[υi+,l−]l−υj−C←[υj−, l]lυi is an o-cycle such that V(C) ⊂ V(C′), a contradiction. Suppose l+υ+j ∈ Ẽ(G). Then C' = PC←[υj, l+]l+υj+C[υj+, υi−]υi−υi+C[υi+,l]lυi is an o-cycle such that V(C) ⊂ V(C′), a contradiction.

(c) Suppose l−l+ ∈ Ẽ(G). Then C′ = PC[υj, υ−i]υ−iυ+iC[υ+i, l−]l−l+C[l+, υ−j]υ−jlυi is an o-cycle such that V(C) ⊂ V(C′), a contradiction. □

Lemma 2.5.

Let G be a 3-connected 1-heavy non-Hamiltonian graph and C be a longest heavy cycle of G, R a component of G − V(C). Let u ∈ V(R), and υ0, υ1, υ2be three neighbors of u which are in the order around C andυ−1, υ+1are light. Letli ∈ C[υ+i, υ−i+1) such thatυ−i+1li ∈ E(G) and li υi ∈ E(G), si ∈ C(υ+i−1, υ−i] such thatυ+i−1si ∈ E(G) and si υi ∈ E(G) (where the indices are taken modulo 3).

(a) Ifυ−2υ+2 ∉ E(G) andυ−0υ+0 ∉ E(G), then (i) υ−1υ2 ∉ E(G), (ii) υ1l−1 ∈ E(G) andυ1s+1 ∈ E(G), (iii) υ1, l+1, l−1, s−1, s+1, l1, s1are light;

(b) Ifυ−2υ+2 ∈ E(G) andυ−0υ+0 ∈ E(G), then {υ+1, l+0, s+2} induces an independent set.

Proof. By Lemma 2.3 (a), uυ−1 ∉ E(G) and uυ+1 ∉ E(G). If υ−1υ+1 ∉ E(G), then since u, υ−1, υ+1 are light, {υ1, u, υ−1, υ+1} induces a light claw, a contradiction. Thus υ−1υ+1 ∈ E(G).

(a) Since υ−2υ+2 ∉ E(G) and υ−0υ+0 ∉ E(G), we have υ−2, υ+0 are heavy or υ+2, υ−0 are heavy by Lemma 2.3 (b).

(i) Suppose υ−1υ2 ∈ E(G) and υ−2, υ+0 are heavy. Let C′ = υ−1υ2C[υ2, υ0]υ2uυ1C[υ1, υ−2]υ−2υ+0C[υ+0, υ−1]. Then C′ is an o-cycle such that V(C) ⊂ V(C′), a contradiction.

Suppose υ−1υ2 ∈ E(G) and υ+2, υ−0 are heavy. Now {υ2, υ−2, u, υ−1} induces a light claw, a contradiction.

(ii) Suppose υ1l−1 ∉ E(G). Note that υ1l+1 ∉ Ẽ(G) and l−1l+1 ∉ Ẽ(G) by Lemma 2.4 (b) and (c). Since υ−2, υ+0 are heavy or υ+2, υ−0 are heavy, by Lemma 2.3 (c) and Lemma 2.4 (b), υ1, l+1, l−1 are light. Now {l1, l+1, υ1, l−1} induces a light claw, a contradiction. Similarly, we can prove that υ1s+1 ∈ E(G).

(iii) By Lemma 2.3 (c) and Lemma 2.4 (b), υ1, l+1, l−1, s−1, s+1 are light. Since υ1l−1 ∈ E(G), we obtain υ+2l1 ∉ Ẽ(G) and υ+0l1 ∉ Ẽ(G) by Lemma 2.4 (b). Note that either υ+0 or υ+2 is a heavy vertex. This implies l1 is a light vertex. The other assertion that s1 is light can be proved similarly.

(b) Since υ0l0 ∈ E(G) and υ2s2 ∈ E(G), υ+1l+0 ∉ Ẽ(G) and υ+1s+2 ∈ Ẽ(G) by Lemma 2.4 (b). Furthermore, we can prove that l+0s+2 ∉ E(G). (Otherwise, C'= υ0uυ2s2C←[s2,l0+]l0+s2+C[s2+,υ2−]υ2−υ2+C[υ2+,υ0−]υ0−υ0+C[υ0+,l0]l0υ0 is an o-cycle such that V(C) ⊂ V(C′), a contradiction.)

Lemma 2.6.

Let G be a non-Hamiltonian almost distance-hereditary graph, C be a longest cycle (a longest heavy cycle) of G, R a component of G − V(C). If there exists a vertex u ∈ V(R) such that NC(u) = {υ1; υ2, … , υr}, then there hold

(a) for any induced (u, υ)-path P, whereυ ∈ N−C(u) orυ ∈ N+C(u), the length of P is at most 3;

(b) ifυ−iυ+i ∈ Ẽ(G), then there exists a vertexli ∈ C[υ+i, υ−i+1) such thatυ−i+1li ∈ E(G) and liυi ∈ E(G), and there exists a vertexsi ∈ C(υ+i−1, υ−1] such thatυ+i−1si ∈ E(G) and si υi ∈ E(G);

(c) ifυ−iυ+i ∈ E(G) andυ−i+1υ+i+1 ∈ E(G), thenυ+i+1li ∈ E(G); and

(d) ififυ−iυ+i ∈ E(G) andυ−i+1υ+i+1 ∈ E(G), then both {li, l−i, υi, υ−i+1} and {li, l+i, υi, υ+i+1} induce claws.

Proof. (a) Suppose there exists an induced (u, υ)-path P such that the length of P is at least 4. It follows that dP (u, υ) ≥ 4, contradicting the fact that dG(u, υ) = 2 and G is almost distance-hereditary.

(b) Let H = G[{u} ∪ V(C[υi, υi+1])] − {υi+1}. Since dG(υ−i+1, u) = 2 and G is almost distance-hereditary, we have dH(υ−i+1, u) ≤ 3. Since υ−iυ+i ∈ Ẽ(G), by Lemma 2.3 (c), we have υiυ−i+1 ∉ E(G) and dH(υ−i+1, u) = 3. It follows that dH(υ−i+1, υi) = 2. So there exists a vertex li ∈ C[υ+i, υ−i+1) such that υ−i+1li ∈ E(G) and li υi ∈ E(G). The other assertion can be proved similarly.

(c) Suppose υ+i+1li ∉ E(G). Let H = G[{υ+i+1, υ−i+1, li, υi, u}]. By Lemma 2.3 (c), υiυ−i+1 ∉ E(G) and υiυ+i+1 ∉ E(G). We can see H is an induced (u, υ+i+1)-path of length 4 in G, contradicting Lemma 2.6 (a).

(d) By Lemma 2.3 (c) and Lemma 2.4 (b), we have υiυ−i+1 ∉ Ẽ(G) and l−iυ−i+1 ∉ Ẽ(G). By Lemma 2.6 (c) and Lemma 2.4 (b), we have υil−i ∉ E(G). So {li, l−i, υi, υ−i+1} induces a claw. The other assertion can be proved similarly. □

3 Proofs of Theorems 1.3 and 1.4

Proof of Theorem 1.3

Let G be a graph satisfying the condition of Theorem 1.3. Let C be a longest cycle of G and assign an orientation to it. Suppose G is not Hamiltonian. Then V(G)\V(C)≠0. Let R be a component of G − C, and A = {υ1, υ2, … , υk} be the set of neighbors of R on C. Since G is 2-connected, there exists a (υi, υj)-path P = υiu1 … ur υj with all internal vertices in R, and υi, υj ∈ A. Choose P such that:

  1. |V(C(υi, υj))| is as small as possible;

  2. |V(P)| is as small as possible subject to (1).

Claim 1

There is no o-cycle C′ in G such that V(C) ⊂ V(C′).

Proof. Otherwise, C′ is an o-cycle such that V(C) ⊂ V(C′). By Lemma 2.2, there exists a cycle containing all vertices in C′ and longer than C, contradicting the choice of C. □

By Lemma 2.3 (e), without loss of generality, assume that υ−iυ+i ∈ E(G).

Claim 2

r = 1, that is, V(P) = {υj, u1, υj}.

Proof. Suppose r ≥ 2. Consider H = G[V(P) ∪ V(C[υi, υj])] − {υj}. Since υ−iυ+i ∈ E(G), υiυ−j ∉ E(G) by Lemma 2.3 (c). Thus dH(υ−j, υi) ≥ 2. By the choice condition of P and Lemma 2.3 (a), we have dP(υi, ur) ≥ 2 and dH(υ−j, ur) = dH(υ−j, υi) + dH(υj, ur) ≥ 4, which yields a contradiction to the fact G is almost distance-hereditary and dG(υ−j, ur) = 2. Hence V(P) = {υi, u1, υj}. □

Claim 3

|V(C[υi, υj])| ≥ 5.

Proof. Suppose |V(C[υi, vj])| = 4 or |V(C[υi, υj])| = 3. This means C[υi, υj] = υiυ+iυ−iυj or C[υi, υj] = υiυ−iυj. Let C′ = υiu1υjυ−jυ+jC[υ+j, υ−i]υ−iυ+iυi or C′ = υiu1υjυ−jυ+jC[υ+j, υi]. Then C′ is an o-cycle such that V(C) ⊂ V(C′) by Lemma 2.3 (d), contradicting Claim 1. □

Recall that υ–iυ+i ∈ E(G). Let H = G[{u1, υ–i} ∪ V(C[υi, υj])] – {υi}. Since dG(υ–i, u1) = 2 and G is almost distance-hereditary, dH(υ–i, u1) ≤ 3. By Lemma 2.3 (c) and (d), we have υ–iυj ∉ E(G). By the choice of P, u1υ ∉ E(G), where υ ∈ C[υ+i, υ–j]. It follows that dH(υ–i, u1) = 3 and dH(υ–i, υj) = 2. By Lemma 2.3 (b), (c) and (d), υ–iυ–j ∉ E(G) and υ+iυj ∉ E(G). Thus there exists a vertex w ∈ C(υ+i, υ–j) such that υ–iw ∈ E(G) and wυj ∈ E(G). Note that w is well-defined.

Claim 4

wυ+j ∉ Ẽ(G).

Proof. Suppose wυ+j ∉ Ẽ(G). By Lemma 2.4 (a), we obtain υ–iw+ ∉ Ẽ(G). Since υ–iw ∈ E(G), we have υjw+ ∉ Ẽ(G) by Lemma 2.4 (b) and by symmetry. Note that υjυ–i ∉ Ẽ(G) by Lemma 2.3 (c). Thus {w, w+, υj, υ–i} induces an o-light claw in G, a contradiction.

Next we will show that {υj, u1, w, υ+j} induces an o-light claw and get a contradiction. Before proving this fact, the following claim is needed.

Claim 5

u1w1 ∉ Ẽ(G).

Proof. First we will show that w–υ–i ∉ Ẽ(G). Since υ–jυ+j ∈ Ẽ(G) and υj w ∈ E(G), we have w–υ–i ∉ Ẽ(G) by Lemma 2.4 (b) and symmetry.

Next we will show that w−υj ∈ Ẽ(G). Suppose not. Consider the subgraph induced by {w, w−, υj, υ–i}. Note that υjυ–i ∉ Ẽ(G) by Lemma 2.3 (c) and w–υ–i ∉ Ẽ(G) by the analysis above. Then {w, w−, υj, υ–i} induces an o-light claw, a contradiction.

Now we will show that u1w ∉ Ẽ(G), since otherwise, C′ = u1wC[w, υ–j]υ–jυ+jC[υ+j, w–]w–υju1 is an o-cycle such that V(C) ⊂ V(C′), contradicting Claim 1. □

By Claims 4, 5 and Lemma 2.3 (a), {υj, u1, w, υ+j} induces an o-light claw, contradicting the fact G is claw-heavy. The proof of Theorem 1.3 is complete. □

Proof of Theorem 1.4.

Let G be a graph satisfying the condition of Theorem 1.4. By Lemma 2.1, there exists a heavy cycle in G. Now choose a longest heavy cycle C of G and assign an orientation to it. Suppose G is not Hamiltonian. Then V(G)\V(C)≠0. Let R be a component of G − C and A = {w1, w2, … , wk} be the set of neighbors of R on C. Since G is 3-connected, for any vertex u of R, there exists a (u, C; 3)-fan F such that F = (u; Q1, Q2, Q3), where Q1 = ux1 … xr1wi, Q2 = uy1 … yr2wj and Q3 = uz1 … zr3wk, and wi, wj, wk are in the order of the orientation of C.

By the choice of C, all internal vertices of F are not heavy. By Lemma 2.3 (b), there is at most one heavy vertex in N+C(R) and at most one heavy vertex in N–C(R). Without loss of generality, assume that w–i, w+i are light. Hence w–iw+i ∈ E(G), otherwise {wi, w–i, w+i, xr1} induces a light claw, contradicting G is 1-heavy.

Claim 1

There exists a (u, C; 3)-fan F such that V(F) = {u, wi, wj, wk}.

Proof. Now we choose the fan F in such a way that:

  1. Q1 = uwi;

  2. |V(C[wi, wj])| is as small as possible subject to (1);

  3. |V(Q2)| is as small as possible subject to (1) and (2);

  4. |V(C[wk, wi])| is as small as possible subject to (1), (2) and (3);

  5. |V(Q3)| is as small as possible subject to (1), (2), (3) and (4).

Since G is 3-connected, for any neighbor of C in R, say u (with uwi ∈ E(G), where wi ∈ V(C)), there are three disjoint paths from u to C. Obviously, we can choose one such path as uwi. Thus (1) is well-defined, and furthermore, the choice condition of F is well-defined.

Claim 1.1.

V(Q2) = {U, wj}.

Proof. Suppose V(Q2)\{u,wj}≠0. Without loss of generality, set y = yr2. Let H = G[V(Q1) ∪ V(Q2) ∪ V(C[wi, wj])] − {wj}. Note that w–iw+i ∈ E(G). By Lemma 2.3 (c), it is easy to see that wi w–j ∉ E(G), so dH(w–j, wi) ≥ 2. In the meantime, the choice condition (2) implies that N(V(Q2)\{wj})∩V(C(wj,wj))=0. This means that dH (w–j, y) = dH (w–j, wi) + dH (wi, y) ≥ 2 + dH (wi, y). Since G is almost distance-hereditary and dG (w–j, y) = 2, we have dH (w–j, y) = 3 and ywi ∈ E(G). Let F′ = (y; Q′1, Q′2, Q′3) such that Q′1 = ywi, Q′2 = ywj and Q′3 = Q2[y, u]Q3[u, wk]. Then F′ is a (y, C;3)-fan satisfying (1), (2) and |V(Q′2)| = 2, contradicting the choice condition (3), a contradiction. □

Claim 1.2.

V (Q3) = {u, wk}.

Proof. Suppose V(Q3)\{u,wk}≠0. Without loss of generality, set z = zr3.

If zwi ∉ E(G), then set H = G[V (Q1) ∪ V (Q3) ∪ V(C[wk, wi])] − {wk}. Since w–iw+i ∈ E(G), we obtain w+kwi ∉ E(G) by Lemma 2.3 (c). This means dH (w+k, wi) ≥ 2. By the choice condition (4), we have N(V(Q3)\{wk})∩V(C(wk,wi))=0, and hence dH (w+k, z) = dH (w+k, wi) + dH (wi, z). Since zwi ∉ E(G), dH (wi, z) ≥ 2 and we get dH (w+k, z) ≥ 4. It yields a contradiction to the fact G is almost distance-hereditary and dG (w+k, z) = 2.

If zwi ∈ E(G), then set H = G[V(C [wi, wj]) ∪ V(Q3 [u, z])] − {wi}. Note that NC(z)∩V(C(wi,wj])=0 (by the choice conditions (2), (5)) and NC(V(Q3)\{z,wk})∩V(C(wi,wj))=0 (by the choice condition (2)). Since dG(w+i, z) = 2 and G is almost distance-hereditary, dH(w+i, z) ≤ 3. But if w+i, wj ∉ E(G), then the distance from z to w+i in H is at least 4, where in such a shortest path, the path Q3 [z, u] contributes at least 1, the path Q2 [u, wj] contributes 1, a contradiction. Thus we have w+iwj ∈ E(G), and hence w–jw+j ∉ Ẽ(G) by Lemma 2.3 (c). Consider the subgraph induced by {wj, w+i, w+j, u}. Since G is 1-heavy and w+i, u are light, w+j is heavy. Now let H = G[{w–i} ∪ V(C[wi, wj]) ∪ V(Q3[u, z])] – {wi}. Similarly, since dG(w–i, z) = 2, we have dH(w–i, z) = 3, and w–i, wj ∈ E(G). Consider the subgraph induced by {wj, w–i, w–j, u}. Similarly, we can see w–j is heavy, and hence w–jw+j ∈ Ẽ(G), a contradiction. Thus V(Q3) = {u, wk}. □

By Claims 1.1 and 1.2, the proof of Claim 1 is complete. □

By Claim 1, there exists a (u, C; 3)-fan F such that V(F)\V(C) = {u}. Suppose that NC (u) = {υ1, υ2, …, υr} (r ≥ 3) and υ1, υ2, …, υr are in the order of the orientation of C. In the following, all the subscripts of υ are taken modulo r, and υ0 = υr.

By Lemma 2.3 (b), there is at most one heavy vertex in N+C(u) and at most one heavy vertex in N–C (u). Since r ≥ 3, we know that there exists υj ∈ NC (u), such that υ–j, υ+j are light, and hence υ–jυ+j ∈ E(G) by the fact G is 1-heavy. Without loss of generality, assume that υ–1υ+1 ∈ E(G) and υ–1, υ+1 are light. By Lemma 2.6 (b), there exists a vertex l1 ∈ C[υ+1, υ–2) such that υ–2l1 ∈ E(G) and l1υ1 ∈ E(G), and there exists a vertex s1 ∈ C(υ+0, υ–1] such that υ+0s1 ∈ E(G) and s1v1 ∈ E(G).

We divide the proof into two cases.

Case 1

υ–2υ+2 ∉ E(G) andυ–0μ+0 ∉ E(G).

Both {υ, υ–2, υ+2, u} and {υ, υ–0, υ+0, u} induce claws. By Lemma 2.3 (b) and the fact G is 1-heavy, υ–2 and υ+0 are heavy or υ+2 and υ–0 are heavy.

Claim 2

υ–1l1 ∈ E(G) andl1υ2 ∈ E(G).

Proof. Suppose υ–1l1 ∉ E(G). Note that uv–1 ∉ Ẽ(G) by Lemma 2.3 (a). By Lemma 2.5 (a), l1 is light. Now {υ1, l1, u, υ–1} induces a light claw, a contradiction.

Suppose l1v2 ∉ E(G). Let H = G[{υ–1, l1, υ–2, υ2, u}]. By Lemma 2.3, we get uv–2 ∉ E(G), uv–1 ∉ E(G) and υ–1υ–2 ∉ E(G). Note that υ2υ–1 ∉ E(G) by Lemma 2.5 (a). Now G[{υ–1, l1, υ–2, υ2, u}] is an induced path of length 4 in G, contradicting Lemma 2.6 (a).

Now we consider the following two subcases.

Subcase 1.1.

υ–2, υ+0are heavy vertices.

By Lemma 2.3 (a), uυ+2 ∉ Ẽ(G). By Lemma 2.5 (a), we have υ1l–1 ∈ E(G). Note that l′ ≔ l–1 ∈ N(υ1) and υ–1υ+1 ∈ E(G). By Lemma 2.4 (b), l′+υ+2 = l1υ+2 ∉ Ẽ(G). By Lemma 2.5 (a) and Lemma 2.3 (b), l1 and υ+2 are light. Now {υ2, l1, u, υ+2} induces a light claw, a contradiction.

Subcase 1.2.

υ+2, υ–0are heavy vertices.

Consider the subgraph induced by {υ+0, s1, l1, υ2, υ}. It is easily to check that υ+0s1 ∈ E(G), l1υ2 ∈ E(G) (by Claim 2) and υ2u ∈ E(G). By Lemma 2.3 (a), υ+0u ∉ E(G). By Lemma 2.5 (a) and Lemma 2.4 (b), we know that υ1l–1 ∈ E(G) and υ+0l1 ∉ E(G). Now we obtain s1l1 ∉ E(G) or υ+0υ2 ∈ E(G) or s1υ2 ∈ E(G) (Otherwise, G[{υ+0, s1, l1, υ2, u}] is an induced path of length 4, contradicting Lemma 2.6 (a)).

Suppose s1l1 ∉ E(G). By Lemma 2.5 (a), l1, s1 are light. Now {υ1, l1, s1, u} induces a light claw, contradicting G is 1-heavy.

Suppose υ+0υ2 ∈ E(G). Consider the subgraph induced by {υ2, υ–2, υ+0, u}. By Lemma 2.3 (a), we have uυ–2 ∉ E(G) and uυ+0 ∉ E(G). Since υ–2, υ+0, u are light and G is 1-heavy, υ+0υ–2 ∈ E(G). By Lemma 2.5 (a) and Claim 2, υ1l–1 ∈ E(G) and l1υ2 ∈ E(G). Now C′=υ1l1−C←[l1−,υ1+]υ1+υ1−C←[υ1−,υ0+]υ0+υ2−C←[υ2−,l1]l1υ2C[υ2,υ0]υ0uυ1 is an o-cycle such that V(C) ⊂ V(C′), a contradiction.

Suppose s1υ2 ∈ E(G). Consider the subgraph induced by {υ2, υ–2, s1, u}. By Lemma 2.5 (a) and Lemma 2.3 (b), s1 and υ–2 are light. Since G is 1-heavy, s1υ–2 ∈ E(G). Now C′=υ1uυ2C[υ2,s1]s1υ2−C←[υ2−,υ1+]υ1+s1+C[s1+,υ1] is an o-cycle such that V(C) ⊂ V(C′), a contradiction. (First, we can prove s1υ+1 ∈ E(G). Otherwise, {υ1, s1, υ+1, u} induces a light claw, a contradiction. Note that υ+0υ+1 ∉ E(G) and υ+0s+1 ∉ E(G) by Lemma 2.3 (b) and Lemma 2.4 (b). Then we obtain υ+1s+1 ∈ E(G) since otherwise {s1, υ+0, υ+1, s+1} induces a light claw, a contradiction.)

Case 2

υ–2υ+2 ∈ E(G) orυ–0υ+0 ∈ E(G).

Without loss of generality (by symmetry), assume that υ–2υ+2 ∈ E(G).

Subcase 2.1.

υ–0υ+0 ∉ E(G).

By Lemma 2.3 (a), uυ–0 ∉ E(G) and uυ+0 ∉ E(G). Now {υ0, υ–0, υ+0, u} induces a claw. Since G is 1-heavy and u is light, υ–0 is heavy or υ+0 is heavy.

Suppose υ–0 is heavy. By Lemma 2.3 (b), (c) and Lemma 2.4 (b), υ–2, υ1 and l–1 are light. By Lemma 2.6 (d), {l1, l–1, υ1, υ–2} induces a light claw, a contradiction.

Suppose υ+0 is heavy. By Lemma 2.3 (b), (c) and Lemma 2.4 (b), we can see υ+2, υ1 and l+1 are light. By Lemma 2.6 (d), {l1, l+1, υ1, υ+2} induces a light claw, a contradiction. (Note that υ–1υ+1 ∈ E(G) and υ–2υ+2 ∈ E(G). By Lemma 6 (c), l1υ+2 ∈ E(G).

Subcase 2.2.

υ–0υ+0 ∈ E(G).

By Lemma 6 (b), there exists a vertex s2 ∈ C(υ+1, υ–2] such that υ+1s2 ∈ E(G) and s2υ2 ∈ E(G).

Claim 3

(i) υ1is heavy, (ii) l–0, l+0, s–2, s+2are light.

Proof. Recall that the definition of l0 occurred in the condition of Lemma 5 before. Let l0 ∈ C[υ+0, υ–1) such that υ–0l0 ∈ E(G) and l0υ0 ∈ E(G).

(i) By Lemma 2.6 (d), each of {l0, l–0, υ0, υ–1} and {l1, l–1, υ1, υ–2} induces a claw. Since G is 1-heavy, at least one vertex of {l–1, υ1, υ–2} is heavy.

Suppose υ–2 is heavy. By Lemma 2.3 (b), (c) and Lemma 2.4 (b), υ–1, υ0 and l–0 are light. Now {l0, l–0, υ0, υ–1} induces a light claw, contradicting G is 1-heavy.

Suppose l–1 is heavy. By Lemma 2.6 (c), υ+2l1 ∈ E(G). By Lemma 2.4 (a) and (b), υ–1l–1 ∉ Ẽ(G) and υ0l–1 ∉ Ẽ(G). This implies that υ0, υ–1 are light. At the same time, we can prove that l0– is light. (Otherwise, C′=υ0uυ2C←[υ2,l1]l1 υ2−C[υ2+,υ0−]υ0−υ0+C[υ0+,l0−]l0−l1−C←[l1−,l0]lυ0 is an o-cycle such that V(C) ⊂ V(C′), a contradiction.) Now {l0, l–0, υ0, υ–1} induces a light claw, contradicting G is 1-heavy.

Note that {l1, l–1, υ1, υ–2} induces a claw and υ–2, l–1 are light. Since G is 1-heavy, υ1 is heavy.

(ii) Note that υ1 is heavy. If l+0 is heavy, then C′ = υ0uυ1l+0C[l+0, υ–1]υ–1υ+1C[υ+1, υ–0]υ–0υ+0C[υ+0, l0]l0υ0 is an o-cycle such that V(C) ⊂ V(C′), a contradiction. If l–0 is heavy, then C′ = υ0l0C[l0, υ–1]υ–1υ+1C[υ+1, υ–0]υ–0υ+0C[υ+0, l–0]l–0υ1uυ0 is an o-cycle such that V(C) ⊂ V(C′), a contradiction. Similarly, by symmetry, we can prove that s–2, s+2 are light. □

Claim 4

υ–1l+0 ∈ E(G) andυ–1s+2 ∈ E(G).

Proof. Suppose υ–1l+0 ∉ E(G). By Lemma 2.4 (b) and (c), υ–1l–0 ∉ Ẽ(G) and l–0l+0 ∉ Ẽ(G). By Claim 3, l+0, l–0 are light. Now {l0, l–0, l+0, υ–1} induces a light claw, a contradiction.

By Lemma 2.6 (c) and by symmetry, we obtain υ–1s2 ∈ E(G). Suppose υ–1s+2 ∉ E(G). By Lemma 2.4 (b) and (c), υ–1s–2 ∉ Ẽ(G) and s–2s+2 ∉ Ẽ(G). By Claim 3, s+2, s–2 are light. Now {s2, s–2, s+2, υ–1} induces a light claw, a contradiction. □

By Claim 3, Lemma 2.5 (b) and Claim 4, l+0, s+2 are light, {υ1+l0+,υ1+s2+,l0+s2+}∩E(G)=0 and {υ–1l+0, υ–1s+2} ⊂ E(G). It is proved that {υ–1, υ+1, l+0, s+0} induces a light claw, a contradiction.

The proof of Theorem 1.4 is complete. □

Acknowledgement

The authors are particularly grateful to the anonymous referee for his/her extensive comments that considerably improved the paper.

The first author is supported by NSFC (No. 11271300) and the Scientific Research Program of Shaanxi Provincial Education Department (No. 2013JK0580).

References

[1] Bondy J.A., Murty, U.S.R., Graph Theory with Applications, Macmillan London and Elsevier, New York, 1976.10.1007/978-1-349-03521-2Search in Google Scholar

[2] Karp R.M., Reducibility among combinatorial problems, In: R.E. Miler et al: Complexity of Computer Computations, Plenum Press, New York, 1972, 85–103.10.1007/978-1-4684-2001-2_9Search in Google Scholar

[3] Howorka E., A characterization of distance-hereditary graphs, Quart. J. Math. Oxford Ser., 1977, 28(112), 417–420.10.1093/qmath/28.4.417Search in Google Scholar

[4] S. Hsieh, C. Ho, T. Hsu, M. Ko, Efficient Algorithms for the Hamiltonian Problem on Distance-Hereditary Graphs, Computing and Combinatorics, Lecture Notes in Computer Science, 2002, 2387, 77–86.10.1007/3-540-45655-4_10Search in Google Scholar

[5] D’Atri A., Moscarini M., Distance-hereditary graphs, Steiner trees, and connected domination, SIAM J. Comput., 1988, 17(3), 521–538.10.1137/0217032Search in Google Scholar

[6] Cogis O., Thierry E., Computing maximum stable sets for distance-hereditary graphs, Discrete Optim., 2005, 2(2), 185–188.10.1016/j.disopt.2005.03.004Search in Google Scholar

[7] Aïder M., Almost distance-hereditary graphs, Discrete Math., 2002, 242(1–3), 1–16.10.1016/S0012-365X(00)00401-5Search in Google Scholar

[8] Feng J.-F., Guo Y.-B., Hamiltonian problem on claw-free and almost distance-hereditary graphs, Discrete Math., 2008, 308(24), 6558–6563.10.1016/j.disc.2007.11.063Search in Google Scholar

[9] Broersma H.J., Ryjáč ek Z., Schiermeyer I., Dirac’s minimum degree condition restricted to claws, Discrete Math., 1997, 167/168, 155–166.10.1016/S0012-365X(96)00224-5Search in Google Scholar

[10] Fujisawa J., Yamashita T., Degree conditions on claws and modified claws for Hamiltonicity of graphs, Discrete Math., 2008, 308(9), 1612–1619.10.1016/j.disc.2007.04.018Search in Google Scholar

[11] Č ada, R., Degree conditions on induced claws, Discrete Math., 2008, 308(23), 5622–5631.10.1016/j.disc.2007.10.026Search in Google Scholar

[12] Feng J.-F., Guo Y.-B., Hamiltonian cycle in almost distance-hereditary graphs with degree condition restricted to claws, Optimazation, 2008, 57(1), 135–141.10.1080/02331930701778981Search in Google Scholar

[13] Chen B., Zhang S.-G., Qiao S.N., Hamilton cycles in claw-heavy graphs, Discrete Math., 2009, 309(8), 2015–2019.10.1016/j.disc.2008.04.013Search in Google Scholar

[14] Ning B., Zhang S.-G., Ore- and Fan-type heavy subgraphs for Hamiltonicity of 2-connected graphs, Discrete Math., 2013, 313(17), 1715–1725.10.1016/j.disc.2013.04.023Search in Google Scholar

[15] Ning B., Zhang, S.-G., Chen, B., Degree and neighborhood intersection conditions restricted to induced subgraphs ensuring Hamiltonicity of graphs, Discrete Math. Algorithms Appl., 2014, 6(3), 1450043, 7pp.10.1142/S1793830914500438Search in Google Scholar

[16] Li B.-L., Ryjáček Z., Wang Y., Zhang S.-G., Pairs of heavy subgraphs for Hamiltonicity of 2-connected graphs, SIAM J. Discrete Math., 2012, 26(3), 1088–1103.10.1137/11084786XSearch in Google Scholar

[17] Bollobás B., Brightwell G., Cycles through specified vertices, Combinatorica, 1993, 13(2), 147–155.10.1007/BF01303200Search in Google Scholar

[18] Shi R.-H., 2-neighborhoods and hamiltonian conditions, J. Graph Theory, 1992, 16(3), 267–271.10.1002/jgt.3190160310Search in Google Scholar

Received: 2015-8-4
Accepted: 2015-10-27
Published Online: 2016-2-9
Published in Print: 2016-1-1

© 2015 Chen and Ning, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Articles in the same Issue

  1. Regular Article
  2. A metric graph satisfying w41=1 that cannot be lifted to a curve satisfying dim⁡ (W41)=1
  3. Regular Article
  4. On the Riemann-Hilbert problem in multiply connected domains
  5. Regular Article
  6. Hamilton cycles in almost distance-hereditary graphs
  7. Regular Article
  8. Locally adequate semigroup algebras
  9. Regular Article
  10. Parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces
  11. Corrigendum
  12. Corrigendum to: parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces
  13. Regular Article
  14. Some new bounds of the minimum eigenvalue for the Hadamard product of an M-matrix and an inverse M-matrix
  15. Regular Article
  16. Integral inequalities involving generalized Erdélyi-Kober fractional integral operators
  17. Regular Article
  18. Results on the deficiencies of some differential-difference polynomials of meromorphic functions
  19. Regular Article
  20. General numerical radius inequalities for matrices of operators
  21. Regular Article
  22. The best uniform quadratic approximation of circular arcs with high accuracy
  23. Regular Article
  24. Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions
  25. Regular Article
  26. A note on the rate of convergence for Chebyshev-Lobatto and Radau systems
  27. Regular Article
  28. On the weakly(α, ψ, ξ)-contractive condition for multi-valued operators in metric spaces and related fixed point results
  29. Regular Article
  30. Existence of a common solution for a system of nonlinear integral equations via fixed point methods in b-metric spaces
  31. Regular Article
  32. Bounds for the Z-eigenpair of general nonnegative tensors
  33. Regular Article
  34. Subsymmetry and asymmetry models for multiway square contingency tables with ordered categories
  35. Regular Article
  36. End-regular and End-orthodox generalized lexicographic products of bipartite graphs
  37. Regular Article
  38. Refinement of the Jensen integral inequality
  39. Regular Article
  40. New iterative codes for 𝓗-tensors and an application
  41. Regular Article
  42. A result for O2-convergence to be topological in posets
  43. Regular Article
  44. A fixed point approach to the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation
  45. Regular Article
  46. Uncertainty orders on the sublinear expectation space
  47. Regular Article
  48. Generalized derivations of Lie triple systems
  49. Regular Article
  50. The BV solution of the parabolic equation with degeneracy on the boundary
  51. Regular Article
  52. Malliavin method for optimal investment in financial markets with memory
  53. Regular Article
  54. Parabolic sublinear operators with rough kernel generated by parabolic calderön-zygmund operators and parabolic local campanato space estimates for their commutators on the parabolic generalized local morrey spaces
  55. Regular Article
  56. On annihilators in BL-algebras
  57. Regular Article
  58. On derivations of quantales
  59. Regular Article
  60. On the closed subfields of Q¯~p
  61. Regular Article
  62. A class of tridiagonal operators associated to some subshifts
  63. Regular Article
  64. Some notes to existence and stability of the positive periodic solutions for a delayed nonlinear differential equations
  65. Regular Article
  66. Weighted fractional differential equations with infinite delay in Banach spaces
  67. Regular Article
  68. Laplace-Stieltjes transform of the system mean lifetime via geometric process model
  69. Regular Article
  70. Various limit theorems for ratios from the uniform distribution
  71. Regular Article
  72. On α-almost Artinian modules
  73. Regular Article
  74. Limit theorems for the weights and the degrees in anN-interactions random graph model
  75. Regular Article
  76. An analysis on the stability of a state dependent delay differential equation
  77. Regular Article
  78. The hybrid mean value of Dedekind sums and two-term exponential sums
  79. Regular Article
  80. New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations
  81. Regular Article
  82. On the concept of general solution for impulsive differential equations of fractional-order q ∈ (2,3)
  83. Regular Article
  84. A Riesz representation theory for completely regular Hausdorff spaces and its applications
  85. Regular Article
  86. Oscillation of impulsive conformable fractional differential equations
  87. Regular Article
  88. Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex
  89. Regular Article
  90. Homoclinic solutions of 2nth-order difference equations containing both advance and retardation
  91. Regular Article
  92. When do L-fuzzy ideals of a ring generate a distributive lattice?
  93. Regular Article
  94. Fully degenerate poly-Bernoulli numbers and polynomials
  95. Commentary
  96. Commentary to: Generalized derivations of Lie triple systems
  97. Regular Article
  98. Simple sufficient conditions for starlikeness and convexity for meromorphic functions
  99. Regular Article
  100. Global stability analysis and control of leptospirosis
  101. Regular Article
  102. Random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise
  103. Regular Article
  104. The fuzzy metric space based on fuzzy measure
  105. Regular Article
  106. A classification of low dimensional multiplicative Hom-Lie superalgebras
  107. Regular Article
  108. Structures of W(2.2) Lie conformal algebra
  109. Regular Article
  110. On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs
  111. Regular Article
  112. Parabolic Marcinkiewicz integrals on product spaces and extrapolation
  113. Regular Article
  114. Prime, weakly prime and almost prime elements in multiplication lattice modules
  115. Regular Article
  116. Pochhammer symbol with negative indices. A new rule for the method of brackets
  117. Regular Article
  118. Outcome space range reduction method for global optimization of sum of affine ratios problem
  119. Regular Article
  120. Factorization theorems for strong maps between matroids of arbitrary cardinality
  121. Regular Article
  122. A convergence analysis of SOR iterative methods for linear systems with weak H-matrices
  123. Regular Article
  124. Existence theory for sequential fractional differential equations with anti-periodic type boundary conditions
  125. Regular Article
  126. Some congruences for 3-component multipartitions
  127. Regular Article
  128. Bound for the largest singular value of nonnegative rectangular tensors
  129. Regular Article
  130. Convolutions of harmonic right half-plane mappings
  131. Regular Article
  132. On homological classification of pomonoids by GP-po-flatness of S-posets
  133. Regular Article
  134. On CSQ-normal subgroups of finite groups
  135. Regular Article
  136. The homogeneous balance of undetermined coefficients method and its application
  137. Regular Article
  138. On the saturated numerical semigroups
  139. Regular Article
  140. The Bruhat rank of a binary symmetric staircase pattern
  141. Regular Article
  142. Fixed point theorems for cyclic contractive mappings via altering distance functions in metric-like spaces
  143. Regular Article
  144. Singularities of lightcone pedals of spacelike curves in Lorentz-Minkowski 3-space
  145. Regular Article
  146. An S-type upper bound for the largest singular value of nonnegative rectangular tensors
  147. Regular Article
  148. Fuzzy ideals of ordered semigroups with fuzzy orderings
  149. Regular Article
  150. On meromorphic functions for sharing two sets and three sets in m-punctured complex plane
  151. Regular Article
  152. An incremental approach to obtaining attribute reduction for dynamic decision systems
  153. Regular Article
  154. Very true operators on MTL-algebras
  155. Regular Article
  156. Value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations
  157. Regular Article
  158. A class of 3-dimensional almost Kenmotsu manifolds with harmonic curvature tensors
  159. Regular Article
  160. Robust dynamic output feedback fault-tolerant control for Takagi-Sugeno fuzzy systems with interval time-varying delay via improved delay partitioning approach
  161. Regular Article
  162. New bounds for the minimum eigenvalue of M-matrices
  163. Regular Article
  164. Semi-quotient mappings and spaces
  165. Regular Article
  166. Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces
  167. Regular Article
  168. A family of singular functions and its relation to harmonic fractal analysis and fuzzy logic
  169. Regular Article
  170. Solution to Fredholm integral inclusions via (F, δb)-contractions
  171. Regular Article
  172. An Ulam stability result on quasi-b-metric-like spaces
  173. Regular Article
  174. On the arrowhead-Fibonacci numbers
  175. Regular Article
  176. Rough semigroups and rough fuzzy semigroups based on fuzzy ideals
  177. Regular Article
  178. The general solution of impulsive systems with Riemann-Liouville fractional derivatives
  179. Regular Article
  180. A remark on local fractional calculus and ordinary derivatives
  181. Regular Article
  182. Elastic Sturmian spirals in the Lorentz-Minkowski plane
  183. Topical Issue: Metaheuristics: Methods and Applications
  184. Bias-variance decomposition in Genetic Programming
  185. Topical Issue: Metaheuristics: Methods and Applications
  186. A novel generalized oppositional biogeography-based optimization algorithm: application to peak to average power ratio reduction in OFDM systems
  187. Special Issue on Recent Developments in Differential Equations
  188. Modeling of vibration for functionally graded beams
  189. Special Issue on Recent Developments in Differential Equations
  190. Decomposition of a second-order linear time-varying differential system as the series connection of two first order commutative pairs
  191. Special Issue on Recent Developments in Differential Equations
  192. Differential equations associated with generalized Bell polynomials and their zeros
  193. Special Issue on Recent Developments in Differential Equations
  194. Differential equations for p, q-Touchard polynomials
  195. Special Issue on Recent Developments in Differential Equations
  196. A new approach to nonlinear singular integral operators depending on three parameters
  197. Special Issue on Recent Developments in Differential Equations
  198. Performance and stochastic stability of the adaptive fading extended Kalman filter with the matrix forgetting factor
  199. Special Issue on Recent Developments in Differential Equations
  200. On new characterization of inextensible flows of space-like curves in de Sitter space
  201. Special Issue on Recent Developments in Differential Equations
  202. Convergence theorems for a family of multivalued nonexpansive mappings in hyperbolic spaces
  203. Special Issue on Recent Developments in Differential Equations
  204. Fractional virus epidemic model on financial networks
  205. Special Issue on Recent Developments in Differential Equations
  206. Reductions and conservation laws for BBM and modified BBM equations
  207. Special Issue on Recent Developments in Differential Equations
  208. Extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances
https://doi.org/10.1515/math-2016-0003
Keywords for this article
Hamilton cycle; Almost distance-hereditary graph; Claw-free graph; 1-heavy graph; 2-heavy graph; Claw-heavy graph
Creative Commons
BY-NC-ND 3.0

Articles in the same Issue

  1. Regular Article
  2. A metric graph satisfying w41=1 that cannot be lifted to a curve satisfying dim⁡ (W41)=1
  3. Regular Article
  4. On the Riemann-Hilbert problem in multiply connected domains
  5. Regular Article
  6. Hamilton cycles in almost distance-hereditary graphs
  7. Regular Article
  8. Locally adequate semigroup algebras
  9. Regular Article
  10. Parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces
  11. Corrigendum
  12. Corrigendum to: parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces
  13. Regular Article
  14. Some new bounds of the minimum eigenvalue for the Hadamard product of an M-matrix and an inverse M-matrix
  15. Regular Article
  16. Integral inequalities involving generalized Erdélyi-Kober fractional integral operators
  17. Regular Article
  18. Results on the deficiencies of some differential-difference polynomials of meromorphic functions
  19. Regular Article
  20. General numerical radius inequalities for matrices of operators
  21. Regular Article
  22. The best uniform quadratic approximation of circular arcs with high accuracy
  23. Regular Article
  24. Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions
  25. Regular Article
  26. A note on the rate of convergence for Chebyshev-Lobatto and Radau systems
  27. Regular Article
  28. On the weakly(α, ψ, ξ)-contractive condition for multi-valued operators in metric spaces and related fixed point results
  29. Regular Article
  30. Existence of a common solution for a system of nonlinear integral equations via fixed point methods in b-metric spaces
  31. Regular Article
  32. Bounds for the Z-eigenpair of general nonnegative tensors
  33. Regular Article
  34. Subsymmetry and asymmetry models for multiway square contingency tables with ordered categories
  35. Regular Article
  36. End-regular and End-orthodox generalized lexicographic products of bipartite graphs
  37. Regular Article
  38. Refinement of the Jensen integral inequality
  39. Regular Article
  40. New iterative codes for 𝓗-tensors and an application
  41. Regular Article
  42. A result for O2-convergence to be topological in posets
  43. Regular Article
  44. A fixed point approach to the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation
  45. Regular Article
  46. Uncertainty orders on the sublinear expectation space
  47. Regular Article
  48. Generalized derivations of Lie triple systems
  49. Regular Article
  50. The BV solution of the parabolic equation with degeneracy on the boundary
  51. Regular Article
  52. Malliavin method for optimal investment in financial markets with memory
  53. Regular Article
  54. Parabolic sublinear operators with rough kernel generated by parabolic calderön-zygmund operators and parabolic local campanato space estimates for their commutators on the parabolic generalized local morrey spaces
  55. Regular Article
  56. On annihilators in BL-algebras
  57. Regular Article
  58. On derivations of quantales
  59. Regular Article
  60. On the closed subfields of Q¯~p
  61. Regular Article
  62. A class of tridiagonal operators associated to some subshifts
  63. Regular Article
  64. Some notes to existence and stability of the positive periodic solutions for a delayed nonlinear differential equations
  65. Regular Article
  66. Weighted fractional differential equations with infinite delay in Banach spaces
  67. Regular Article
  68. Laplace-Stieltjes transform of the system mean lifetime via geometric process model
  69. Regular Article
  70. Various limit theorems for ratios from the uniform distribution
  71. Regular Article
  72. On α-almost Artinian modules
  73. Regular Article
  74. Limit theorems for the weights and the degrees in anN-interactions random graph model
  75. Regular Article
  76. An analysis on the stability of a state dependent delay differential equation
  77. Regular Article
  78. The hybrid mean value of Dedekind sums and two-term exponential sums
  79. Regular Article
  80. New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations
  81. Regular Article
  82. On the concept of general solution for impulsive differential equations of fractional-order q ∈ (2,3)
  83. Regular Article
  84. A Riesz representation theory for completely regular Hausdorff spaces and its applications
  85. Regular Article
  86. Oscillation of impulsive conformable fractional differential equations
  87. Regular Article
  88. Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex
  89. Regular Article
  90. Homoclinic solutions of 2nth-order difference equations containing both advance and retardation
  91. Regular Article
  92. When do L-fuzzy ideals of a ring generate a distributive lattice?
  93. Regular Article
  94. Fully degenerate poly-Bernoulli numbers and polynomials
  95. Commentary
  96. Commentary to: Generalized derivations of Lie triple systems
  97. Regular Article
  98. Simple sufficient conditions for starlikeness and convexity for meromorphic functions
  99. Regular Article
  100. Global stability analysis and control of leptospirosis
  101. Regular Article
  102. Random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise
  103. Regular Article
  104. The fuzzy metric space based on fuzzy measure
  105. Regular Article
  106. A classification of low dimensional multiplicative Hom-Lie superalgebras
  107. Regular Article
  108. Structures of W(2.2) Lie conformal algebra
  109. Regular Article
  110. On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs
  111. Regular Article
  112. Parabolic Marcinkiewicz integrals on product spaces and extrapolation
  113. Regular Article
  114. Prime, weakly prime and almost prime elements in multiplication lattice modules
  115. Regular Article
  116. Pochhammer symbol with negative indices. A new rule for the method of brackets
  117. Regular Article
  118. Outcome space range reduction method for global optimization of sum of affine ratios problem
  119. Regular Article
  120. Factorization theorems for strong maps between matroids of arbitrary cardinality
  121. Regular Article
  122. A convergence analysis of SOR iterative methods for linear systems with weak H-matrices
  123. Regular Article
  124. Existence theory for sequential fractional differential equations with anti-periodic type boundary conditions
  125. Regular Article
  126. Some congruences for 3-component multipartitions
  127. Regular Article
  128. Bound for the largest singular value of nonnegative rectangular tensors
  129. Regular Article
  130. Convolutions of harmonic right half-plane mappings
  131. Regular Article
  132. On homological classification of pomonoids by GP-po-flatness of S-posets
  133. Regular Article
  134. On CSQ-normal subgroups of finite groups
  135. Regular Article
  136. The homogeneous balance of undetermined coefficients method and its application
  137. Regular Article
  138. On the saturated numerical semigroups
  139. Regular Article
  140. The Bruhat rank of a binary symmetric staircase pattern
  141. Regular Article
  142. Fixed point theorems for cyclic contractive mappings via altering distance functions in metric-like spaces
  143. Regular Article
  144. Singularities of lightcone pedals of spacelike curves in Lorentz-Minkowski 3-space
  145. Regular Article
  146. An S-type upper bound for the largest singular value of nonnegative rectangular tensors
  147. Regular Article
  148. Fuzzy ideals of ordered semigroups with fuzzy orderings
  149. Regular Article
  150. On meromorphic functions for sharing two sets and three sets in m-punctured complex plane
  151. Regular Article
  152. An incremental approach to obtaining attribute reduction for dynamic decision systems
  153. Regular Article
  154. Very true operators on MTL-algebras
  155. Regular Article
  156. Value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations
  157. Regular Article
  158. A class of 3-dimensional almost Kenmotsu manifolds with harmonic curvature tensors
  159. Regular Article
  160. Robust dynamic output feedback fault-tolerant control for Takagi-Sugeno fuzzy systems with interval time-varying delay via improved delay partitioning approach
  161. Regular Article
  162. New bounds for the minimum eigenvalue of M-matrices
  163. Regular Article
  164. Semi-quotient mappings and spaces
  165. Regular Article
  166. Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces
  167. Regular Article
  168. A family of singular functions and its relation to harmonic fractal analysis and fuzzy logic
  169. Regular Article
  170. Solution to Fredholm integral inclusions via (F, δb)-contractions
  171. Regular Article
  172. An Ulam stability result on quasi-b-metric-like spaces
  173. Regular Article
  174. On the arrowhead-Fibonacci numbers
  175. Regular Article
  176. Rough semigroups and rough fuzzy semigroups based on fuzzy ideals
  177. Regular Article
  178. The general solution of impulsive systems with Riemann-Liouville fractional derivatives
  179. Regular Article
  180. A remark on local fractional calculus and ordinary derivatives
  181. Regular Article
  182. Elastic Sturmian spirals in the Lorentz-Minkowski plane
  183. Topical Issue: Metaheuristics: Methods and Applications
  184. Bias-variance decomposition in Genetic Programming
  185. Topical Issue: Metaheuristics: Methods and Applications
  186. A novel generalized oppositional biogeography-based optimization algorithm: application to peak to average power ratio reduction in OFDM systems
  187. Special Issue on Recent Developments in Differential Equations
  188. Modeling of vibration for functionally graded beams
  189. Special Issue on Recent Developments in Differential Equations
  190. Decomposition of a second-order linear time-varying differential system as the series connection of two first order commutative pairs
  191. Special Issue on Recent Developments in Differential Equations
  192. Differential equations associated with generalized Bell polynomials and their zeros
  193. Special Issue on Recent Developments in Differential Equations
  194. Differential equations for p, q-Touchard polynomials
  195. Special Issue on Recent Developments in Differential Equations
  196. A new approach to nonlinear singular integral operators depending on three parameters
  197. Special Issue on Recent Developments in Differential Equations
  198. Performance and stochastic stability of the adaptive fading extended Kalman filter with the matrix forgetting factor
  199. Special Issue on Recent Developments in Differential Equations
  200. On new characterization of inextensible flows of space-like curves in de Sitter space
  201. Special Issue on Recent Developments in Differential Equations
  202. Convergence theorems for a family of multivalued nonexpansive mappings in hyperbolic spaces
  203. Special Issue on Recent Developments in Differential Equations
  204. Fractional virus epidemic model on financial networks
  205. Special Issue on Recent Developments in Differential Equations
  206. Reductions and conservation laws for BBM and modified BBM equations
  207. Special Issue on Recent Developments in Differential Equations
  208. Extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 9.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2016-0003/html?lang=en
Scroll to top button