Some lower bounds for maximum colored cuts

1 Introduction

The Max-Cut problem is a classical partition problem: Given a graph G(V, E), V = V(G) is the vertex set of G, E = E(G) is the edge set of G, the objective is to find a bipartition (A, B) of V(G) which maximizes the number of edges joining A and B. This problem has been widely studied in graph theory and combinatorics in recent decades. Let mc(G) denote the maximum number of edges in a cut of G. A basic result is that mc(G) ≥ m/2 (m = |E(G)|), which can be seen by considering the expected value of crossing edges in a uniformly random bipartition. There have been many improvements of this bound. In 1973, Edwards [1],2] showed that

and the bound is tight as evidenced by complete graphs with odd order. For more papers on max-cuts of graphs, we refer readers to [3], [4], [5], [6], [7], [8], [9].

The Maximum Colored Cut problem is a more general problem: Let p be a positive integer and let [p] ≔ {1, 2, …, p}. Given a simple graph G(V, E, c), where c: E → [p] is an edge coloring of G, and for each i ∈ [p], c ⁻¹(i) ≠ ∅, the objective is to find a bipartition (A, B) of V(G) which maximizes the number of colors of edges joining A and B. Obviously, the problem is the classical Max-Cut problem if the number of edges is equal to the number of colors. Faria et al. [10] proposed this problem and studied its complexity. To state conveniently, we first introduce some definitions and notations. Throughout this paper, all graphs considered are finite and simple. For an edge-colored graph G, let col(G) be the set of colors appearing on E(G). We say that G is rainbow if |col(G)| = |E(G)|; and G is properly colored (PC) if each pair of adjacent edges in G are assigned different colors. For any two disjoint subsets S and T of V(G), we use col(S, T) to denote the set of colors appearing on the edges between S and T. If S = {v}, then we write col(v, T) instead of col({v}, T). We also write col(S) instead of col(G[S]), where G[S] is the subgraph of G induced by S. Let mcc(G) denote the maximum number of colors in a cut of G. Obviously, mcc(G) = mc(G) if p = |E(G)| and mcc(G) is close to mc(G) if p is close to |E(G)|. Therefore, we are more interested in the cases where there is a significant difference in the number of edges and colors. Naturally, this can be ensured when the given edge-colored graph does not contain certain given edge-colored subgraphs.

Let K _n be a complete graph with n vertices, and a rainbow K ₃ is referred to as a rainbow triangle. Let C ₄ denote a cycle of length 4. An edge-colored graph G is referred to as a p-edge-colored graph if G contains exactly p distinct colors on its edges. In [11], we began to study the maximum colored cuts in edge-colored complete graphs which contain no rainbow triangles and PC-C ₄s. In [12], we improved the results from [11] and showed that for a p-edge-colored complete graph G, (1) if G contains no rainbow triangles, then mcc(G) = p; (2) if G contains no PC-C ₄s, then mcc(G) ≥ p − 1. In this paper, we continue to explore the maximum colored cuts in edge-colored complete graphs containing no such properly colored subgraph. Let K 4 − be a graph obtained from K ₄ by deleting an edge. For an edge-colored complete graph containing no PC- K 4 − s, we provide the following result.

It should be noted that the lower bound established in Theorem 1.1 is nearly optimal. Let G be a complete graph with |V(G)| = 3l (l ≥ 1). Then G contains l vertex-disjoint triangles. Color the edges of these triangles such that any two edges are assigned different colors, and the remaining edges in G are colored with the (3l + 1)th color. Obviously, such an edge-colored complete graph G contains no PC- K 4 − s and p = 3l + 1. Let (A, B) be a partition such that |col(A, B)| = mcc(G). Then there are at least l colors which do not belong to col(A, B). Hence, mcc(G) ≤ 2l + 1 = 2(p − 1)/3 + 1 = (2p + 1)/3.

Let k ≥ 3 be an integer. A complete k-partite graph is a graph whose vertices can be partitioned into k disjoint sets, known as partite sets such that: (a) There are no edges connecting any two vertices within the same partite set; (b) Each vertex is connected by an edge to each vertex in all the other partite sets. For a p-edge colored complete k-partite graph G, in [12], we showed that mcc(G) ≥ min{p − 1, 15p/16} if G contains no PC-C ₄s. In this paper, we improve this result and give the following theorem.

In the remainder of this paper, we present a proof of Theorem 1.1 in Section 2 and a proof of Theorem 1.2 in Section 3. Finally, we offer concluding remarks and propose several interesting problems.

2 Proof of theorem 1.1

Let p ≥ 4 be an integer and let G be a p-edge-colored complete graph containing no PC- K 4 − s. Let col(G) ≔ [p]. For each i ∈ [p], let G ⁱ be the subgraph of G induced by the edges of color i. Let Δ(G ⁱ ) be the maximum degree of G ⁱ and let

Based on the value of |I|, we proceed with this proof in the following two cases.

Case 1. |I| ≤ p/3.

Let V(G) = U ₁ ∪ U ₂ be a random partition of V(G) by placing each v ∈ V(G) into U ₁ or U ₂, independently, with probability 1/2. For each color i ∈ [p], let X _i be the indicator random variable of the event that there exists an edge e = uv colored i such that either u ∈ U ₁, v ∈ U ₂ or v ∈ U ₁, u ∈ U ₂. For each i ∈ I,

For each i ∈ [p]\I,

It follows that

This indicates that there exists a partition (U ₁, U ₂) of V(G) such that mcc(G) ≥ |col(U ₁, U ₂)|≥ 2p/3.

Case 2. |I| > p/3.

Let H be the subgraph of G induced by ⋃ _i∈I E(G ⁱ ).

Let K _m,n represent a complete bipartite graph, where m and n represent the number of vertices in each of the two sets. By Claim 2.1, each component of H can be a K ₂, a K _1,t (t ≥ 2) or a triangle. Let A 1 be the set that contains all components of H that are isomorphic to K ₂; let A 2 be the set that contains all components of H that are isomorphic to K _1,t (t ≥ 2); and let A 3 be the set that contains all components of H that are isomorphic to K ₃. Let c o l ( A i ) (1 ≤ i ≤ 3) be the color set that contains all colors of all components in A i . For any F ∈ A 2 , let G[V(F)] be the subgraph of G induced by V(F).

Let G′ = G[V(H)] be the subgraph of G induced by V(H) and let |col(G′)| = p′.

If p′ = p, then we are done. So we suppose that T = [p]\ col(G′) ≠ ∅. Let S = V(G)\V(H). Then S ≠ ∅. For each i ∈ T, either Δ(G ⁱ ) ≥ 2 or Δ(G ⁱ ) = 1 and E ( G i ) ≥ 2 . Let

and

Split each vertex in S into V ₁ or V ₂, randomly and independently, such that W ₁ ⊆ V ₁ and W ₂ ⊆ V ₂, where (W ₁, W ₂) is the partition of G′ obtained in the proof of Claim 2.3. For each color k ∈ T, let Z _k be the indicator random variable of the event that k ∈ col(V ₁, V ₂). For each k ∈ T ₁, there exist at least 2 edges in G colored k, and each of them may belong to G[S] or be located between V(H) and S. No matter where they are located,

For each r ∈ T ₂, there exists a vertex u ∈ V(G) such that uv and uw are colored r. If u ∈ V(H), then v, w ∈ S. Thus,

So we suppose that u ∈ S. If v ∈ S or w ∈ S, then (2) also holds. Now we suppose that v, w ∈ V(H). If v ∈ W ₁ and w ∈ W ₂, then r ∈ col(V ₁, V ₂). Hence, we may suppose that v, w ∈ W ₁ and r ∉ col(u, W ₂). Let F be any a component of A 2 , and let z be the central vertex of F.

By Claim 2.4, v and w are randomly and independently assigned to W ₁ or W ₂. Thus, (2) also holds. It follows that

This indicates there exists a partition (V ₁, V ₂) of G such that mcc(G) ≥ |col(V ₁, V ₂)| ≥ 2p/3. This way, the proof of Theorem 1.1 is now complete.

3 Proof of theorem 1.2

Let k ≥ 3 be an integer and let G be a p-edge-colored complete k-partite graph containing no PC-C ₄s. Let col(G) ≔ [p]. Theorem 1.2 is trivial if p ≤ 3. So we suppose that p ≥ 4. Assume by a contradiction that mcc(G) ≤ p − 2. Let (V ₁, V ₂) be a partition of G such that m c c ( G ) = c o l ( V 1 , V 2 ) ≤ p − 2 . That is, there exist at least two colors, say 1, 2 ∈ [p], such that 1, 2 ∉ col(V ₁, V ₂).

Let e ₁ = uv and e ₂ = vw. Let E(V _i ) (i = 1, 2) be the edge set that contains all edges in V _i . Then by symmetry, suppose that e ₁, e ₂ ∈ E(V ₁). Using the same proof as Claim 3.1, we know that 1, 2 ∉ col(V ₂). Let

and let C = V ₂\(A ∪ B). Since G contains no PC-C ₄s, for each x ∈ C, c(ux) = c(wx). If A = B, then c o l ( V 1 \ { u } , V 2 ∪ { u } ) > c o l ( V 1 , V 2 ) , which contradicts the assumption that m c c ( G ) = c o l ( V 1 , V 2 ) . Hence, A ≠ B. By the definition of A and B, A ∩ B = ∅. In other words, u and w are not in the same partite set. Now we claim that A ≠ ∅ and B ≠ ∅. Suppose for a contradiction that A = ∅. Then c o l ( V 1 \ { u } , V 2 ∪ { u } ) > c o l ( V 1 , V 2 ) , a contradiction. Similarly, we can prove that B ≠ ∅.

If for each z ∈ A, c(uz) = c(vz), then c o l ( V 1 \ { u } , V 2 ∪ { u } ) > c o l ( V 1 , V 2 ) . So we suppose that A′ ≠ ∅, where A′ = {z ∈ A|c(uz) ≠ c(vz)}. If col(u, A′) = {c(uw)}, then c o l ( V 1 \ { u } , V 2 ∪ { u } ) > c o l ( V 1 , V 2 ) . So we suppose that there exists a vertex, say z′ ∈ A′, such that c(uz′) ≠ c(uw). Note that c(uz′) ≠ c(vz′) as z′ ∈ A′. Since uz′vwu is not a PC-C ₄, c(uw) = c(vw) = 2. This way, e ₁ = vu, e ₃ = uw are two adjacent edges such that c(e ₁) = 1 and c(e ₃) = 2. Hence, for each x ∈ B, c(vx) = c(wx). For otherwise, xvuwx would be a PC-C ₄. Thus, c o l ( V 1 \ { w } , V 2 ∪ { w } ) > c o l ( V 1 , V 2 ) , a contradiction. This way, we have proven that mcc(G) ≥ p − 1.

Now we show that the bound mcc(G) ≥ p − 1 is tight. That is, we need to construct a p-edge-colored complete k-partite graph G containing no PC-C ₄s, such that mcc(G) = p − 1. Let G be a complete k-partite graph with partite sets U ₁, U ₂, …, U _k . Let U ₁ = {u ₁}, U ₂ = {u ₂} and U ₃ = {u ₃}. Color G′ = G[u ₁, u ₂, u ₃] such that G′ is a rainbow triangle, and color the remaining edges of G with the fourth color. Clearly, such an edge-colored complete k-partite graph G contains no PC-C ₄s and mcc(G) = p − 1. This way, the proof of Theorem 1.2 is now complete.

4 Concluding remarks

In this paper, we provide two lower bounds on the maximum colored cut, and both results are almost optimal. Now we present some concluding remarks. First, we remark that Theorem 1.1 also holds when p < 4. However, to avoid the rainbow triangle, the most natural extremal graph, we restrict our consideration to the case where p ≥ 4. Second, in [12], we showed that mcc(G) ≥ min{15p/16, p − 1} if G is a p-edge-colored complete k-partite graph containing no PC-C ₄s. In fact, in that paper, we primarily proved that mcc(G) ≥ 15p/16. Since 15p/16 < p − 1 when p > 16, Theorem 1.2 improves upon this result. Below, we provide several potential directions for future research. A partition (V ₁, V ₂) of a graph is called a bisection if −1 ≤ |V ₁| − |V ₂| ≤ 1. It is an interesting and more challenging direction to replace bipartition with bisection. It is also interesting to replace PC subgraph with rainbow subgraph for forbidden edge-colored subgraph to determine the bound of mcc(G).