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Topics on data transmission problem in software definition network

  • Wei Gao EMAIL logo , Li Liang , Tianwei Xu and Jianhou Gan
Published/Copyright: August 3, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

Abstract

In normal computer networks, the data transmission between two sites go through the shortest path between two corresponding vertices. However, in the setting of software definition network (SDN), it should monitor the network traffic flow in each site and channel timely, and the data transmission path between two sites in SDN should consider the congestion in current networks. Hence, the difference of available data transmission theory between normal computer network and software definition network is that we should consider the prohibit graph structures in SDN, and these forbidden subgraphs represent the sites and channels in which data can’t be passed by the serious congestion. Inspired by theoretical analysis of an available data transmission in SDN, we consider some computational problems from the perspective of the graph theory. Several results determined in the paper imply the sufficient conditions of data transmission in SDN in the various graph settings.

Keywords: software definition network; data transmission; fractional factor; toughness; Hamiltonian fractional factors
PACS: 02.10.Ox; 07.05.Mh

1 Introduction

We equate the fractional factor problem with a relaxation of the famous cardinality matching problem which is one of the core problems in operation research. It has widely applications in various fields such as network design, combinatorial polyhedron and scheduling. For example, several large data packets are sent to different destinations through some channels in a data transmission network. This work helps to partition the large data packets into small ones and then make it efficiency improved. The available allocations of data packets is equated with the problem of fractional flow, which converts to fractional factor problem in a graph generated by a network.

Specifically, there is a graph that can model the complete network and the graph needs to meet the requirements that each site corresponds to a vertex and each channel corresponds to an edge in it. In normal network, the path of data transmission is selected by the shortest path between vertices. Several contributions on data transmission in networks are presented recently. Rolim et al. [1] studied urban sensing problem by means of opportunistic networks to support the data transmission. Vahidi et al. [2] provided the high-mobility airborne hyperspectral data transmission algorithm in view of unmanned aerial vehicles approach. Miridakis et al. [3] determined a rather cost-effective solution for the dual-hop cognitive secondary relaying system. Lee et al. [4] considered streaming data transmission on a discrete memoryless channel. However, in the setting of software definition network, the data transmission depends on the network flow computation. The transmission path is selected with minimum transmission congestion in the current moment. From this perspective, the model of data transmission problem in SDN is becoming what makes the fractional factor avoid certain subgraphs.

It is only the simple graphs that are chosen in our article. Let G = (V(G), E(G)) be a graph, in which V(G) and E(G) are the vertex set and the edge set, respectively. Let n = |V(G)| be the order of graph G. The standard notations and terminologies used but undefined clearly in this article can be found in Bondy and Mutry [5], Basavanagoud et al. [6], Wang [7], and Gao and Wang [8]. The toughness t(G) of a graph G can be stated in the below:

t(G)=min{|S|ω(G−S)|S⊆V(G),ω(G−S)≥2},

if G is not complete; otherwise, t(G) = +∞. It is an important parameter, which is used to measure the vulnerability of networks.

Let g and f be two positive integer-valued functions on V(G) such that 0 ≤ g(x) ≤ f(x) for any x ∈ V(G). A fractional (g, f)-factor is a function h which assigns a number in [0,1] to each edge so that g(x)≤dGh(x)≤f(x) for each x ∈ V(G), where dGh(x)=∑e∈E(x)h(e) is called the fractionaldegree of x in G. A fractional (a, b)-factor (here a and b are both positive integers and a ≤ b) is a function h that assigned a number [0,1] to each edge of a graph G so that for each vertex x we have a≤dGh(x)≤b. Clearly, fractional (a, b)-factor is a particular case of fractional (g, f)-factor when g(x) = a and f(x) = b for any x ∈ V(G). If a = b = k(k ≥ 1 is an integer) for all x ∈ V(G), then a fractional (a, b)-factor is just a fractional k-factor.

It’s stated that G includes a Hamiltonian fractional (g, f)-factor if G has a fractional (g, f)-factor containing a Hamiltonian cycle. A graph G is said to be an ID- Hamiltonian graph if the remaining graph of G admits a Hamiltonian cycle, when we delete any independent set of G. We say that G has an ID-Hamiltonian fractional (g, f)-factor if the remaining graph of G includes a Hamiltonian fractional (g, f)-factor when we delete any independent set of G.

In SDN, the independent set I is corresponding to the website with high transmission congestion and a Hamiltonian cycle corresponding to several channels with high transmission congestion. Thus, we study the data transmission problem by considering the Hamiltonian fractional (g, f)-factor and ID-Hamiltonian fractional (g, f)- factor in the networks model. Several related results can refer to [9-14].

The sufficient conditions of Hamiltonian fractional (g, f)-factors and ID-Hamiltonian fractional (g, f)-factors in graphs are studied and at last we obtain three results that are stated in the below.

Theorem 1

Let a, b be two integers with 3 ≤ a ≤ b, and G be a Hamiltonian graph of order n>(a+b−5)(a+b−3)a−2.g,f are taken to be two integer-valued functions that are defined on V(G) such that a ≤ g(x) ≤ f(x) ≤ b for each x ∈ V(G). If

δ(G)≥(b−1)n+a+b−3a+b−3

and

δ(G)>(b−2)n+2α(G)−1a+b−4,

then G has a Hamiltonian fractional (g, f)-factor.

Theorem 2

Let a, b be two integers with 3 ≤ a ≤ b, and G be a Hamiltonian graph of order n ≥ b + 2. g, f are taken to be two integer-valued functions that are defined on V(G) such that a ≤ g(x) ≤ f(x) ≤ b for each x ∈ V(G). If t(G) ≥ b − 1+b−1a−2, then G has a Hamiltonian fractional (g, f )-factor.

Theorem 3

a, b are taken to be two integers with 3 ≤ a ≤ b, and G be an ID-Hamiltonian graph. And g, f are taken to be two integer-valued functions that are defined on V(G) such that a ≤ g(x) ≤ f(x) ≤ b for each x ∈ V(G). If

κ(G)≥max(b+1)2+4(a−2)2,(a+b+1)(a+b−3)4(a−2)+(a+b+1)(a+b−3)(b+1)2(1)

and

κ(G)≥(b+1)2+4(a−2)4(a−2)α(G),

then G has an ID-Hamiltonian fractional (g, f)-factor.

The ways to prove our major results are on the basis of the listed lemmas below:

Lemma 1

(Liu and Zhang [15]) Since G is a graph and we have H = G[T], so δ(H) ≥ 1 and 1 ≤ dG(x) ≤ k − 1 for every x ∈ V(H) where T ⊆ V(G) and k ≥ 2. Let T1,..., Tk-1 be a partition of the vertices of H satisfying dG(x) = j for each x ∈ Tj where some Tj are allowed to be empty on purpose. If each component of H has a vertex of degree at most k − 2 in G, then H has a maximal independent set I and a covering set C = V(H) − I such that

∑j=1k−1(k−j)cj≤∑j=1k−1(k−2)(k−j)ij,

where cj = |C ∩ Tj | and ij = |I ∩ Tj| for every j = 1,..., k − 1.

Lemma 2

(Liu and Zhang [15]) Since G is a graph and H = G[T], so dG(x) = k − 1 for every x ∈ V(H) and no component of H is isomorphic to Kk where T ⊆ V(G) and k ≥ 2. Then a maximal independent set I is obtained and the covering set C = V(H) − I of H satisfy

|V(H)|≤(k−1k+1)|I|,

and

|C|≤(k−1−1k+1)|I|.

Lemma 3

(Anstee [16]) Assume f and g be two integervalued functions that are defined on the vertex set of a graph G such that 0 ≤ g(x) ≤ f(x) for each x ∈ V(G). Then G has a fractional (g, f)-factor if and only if for every subset S of V(G), g(T) − dG-S(T) ≤ f(S),where T = {x ∈ V(G)\S | dG-S(x) ≤ g(x) − 1}.

Clearly, Lemma 3 is equal to the following version.

Lemma 4

Suppose that f and g are two integer-valued functions that are defined on the vertex set of a graph G such that 0 ≤ g(x) ≤ f(x) for each x ∈ V(G). Then G has a fractional (g, f )-factor if and only if

f(S)+dG−S(T)−g(T)≥0

for all disjoint subsets S, T of V(G).

Lemma 5

(Katerinis [17]) If the graph G isn’t complete, then t(G)≤δ(G)2.

The tricks used to prove our main results are borrowed from Zhou et al. [18], [19] and [20], and more new technologies are introduced here to deal with complex problems.

2 Proof of Theorem 1

In terms of the condition of Theorem 1 and the definition of Hamiltonian graph, G admits a Hamiltonian cycle C. Let G′ = G − E(C). Clearly, G includes the desired fractional factor if H contains a fractional (g − 2, f − 2)-factor. In view of contradiction, we assume that G′ has no fractional (g − 2, f − 2)-factor. Hence, concerning Lemma 4, there are some disjoint subset S and T of V(H) satisfying

(a−2)|S|+dG′−S(T)−(b−2)|T|≤(f−2)(S)+dG′−S(T)−(g−2)(T)≤−1.(2)

We select suitable subsets S and T with the smallest |T|.

If T = ∅, then using (2) we obtain −1 ≥ (a − 2)|S| ≥ 0, a contradiction. Thus, T ≠ ∅. Define h = min{dR-S(x) : x ∈ T}. We infer,

δ(G)≤dG(x1)≤dG−S(x1)+|S|=h+|S|.(3)

Now, the following facts are to be concerned.

Claim 1.1

dG′-S(x) ≤ b − 3 for any x ∈ T.

Proof

If dG′-S(x) ≥ b − 2 for certain x ∈ T, then the subset S and T − {x} satisfy (2) as well. This is contradictive with the selection of T. □

Claim 1.2

dG-S(x) ≤ dG′-S(x) + 2 ≤ b − 1 for all x ∈ T.

Proof

From Claim 1.1 and G′ = G − E(C), we deduce

dG−S(x)≤dG′−S(x)+2≤b−1

for any x ∈ T. □

Taking the definition of h and Claim 1.2 into consideration, we yield 0 ≤ h ≤ b − 1. The following proof is composed of two cases in terms of the value of h.

Case 1

h = 0.

Set X = {x ∈ T : dG-S(x) = 0}, Y = {x ∈ T : dG-S(x) = 1}, Y1 = {x ∈ Y : NG-S(x) ⊆ T} and Y2 = Y − Y1· G[Y1] has maximum degree at most 1 in G − S. Let Z be a maximum independent set of G[Y1]. We get |Z|≥12|Y1|. On the basis of definitions, X ∩ Z ∩ Y2 is an independent set of G. Therefore, we get

α(G)≥|X|+|Z|+|Y2|≥|X|+12|Y1|+12|Y2|=|X|+12|Y|.(4)

In terms of (2), (3), (4) and Claim 1.2, we infer

α(G)−1≥|X|+12|Y|+(a−2)|S|+dG′−S(T)−(b−2)|T|≥|X|+12|Y|+(a−2)|S|+dG−S(T)−2|T|−(b−2)|T|=|X|+12|Y|+(a−2)|S|+dG−S(T)−b|T|=|X|+12|Y|+(a−2)|S|+dG−S(T−X∪Y)+|Y|−b|T|=|X|+32|Y|+(a−2)|S|+dG−S(T−X∪Y)−b|T|≥|X|+32|Y|+(a−2)|S|+2|T−X∪Y|−b|T|=(a−2)|S|−(b−2)|T|−(|X|+12|Y|)≥(a−2)δ(G)−(b−2)|T|−α(G),
i.e.
(b−2)|T|≥(a−2)δ(G)−2α(G)+1.(5)

For b ≥ 3, (3), (5) and |S| + |T| ≤ n, we get

0≤n−|S|−|T|≤n−δ(G)−(a−2)δ(G)−2α(G)+1b−2=(b−2)n−(a+b−4)δ(G)+2α(G)−1b−2,

that is

δ(G)≤(b−2)n+2α(G)−1a+b−4.

This contradicts δ(G)>(b−2)n+2α(G)−1a+b−4.

Case 2

1 ≤ h ≤ b − 1.

In light of (2) and |S| + |T| ≤ n, we derive

−1≥(a−2)|S|+dG′−S(T)−(b−2)|T|≥(a−2)|S|+dG−S(T)−2|T|−(b−2)|T|=(a−2)|S|+dG−S(T)−b|T|≥(a−2)|S|+h|T|−b|T|=(a−2)|S|−(b−h)|T|≥(a−2)|S|−(b−h)(n−|S|)=(a+b−h−2)|S|−(b−h)n,

which implies

|S|≤(b−h)n−1a+b−h−2.(6)

Using (3) and (6), we deduce

δ(G)≤|S|+h≤(b−h)n−1a+b−h−2+h.(7)

Subcase 2.1

h = 1.

By means of (7), we yield

δ(G)≤(b−1)n−1a+b−3+1=(b−1)n+a+b−4a+b−3,

which contradicts δ(G)≥(b−1)n+a+b−3a+b−3.

Subcase 2.2

2 ≤ h = b − 1.

Set f(h)=(b−h)n−1a+b−h−2+h. From 2 ≤ h ≤ b − 1 and n > (a+b−5)(a+b−3)a−2, we get

f′(h)=−n(a+b−h−2)+(b−h)n−1(a+b−h−2)2+1=−n(a−2)−1(a+b−h−2)2+1≤−n(a−2)−1(a+b−4)2+1<−(a+b−5)(a+b−3)−1(a+b−4)2+1=0.

Hence, max{f(h)} = f(2). According to (7) and n > (a+b−5)(a+b−3)a−2, we have

δ(G)≤f(h)≤f(2)=(b−2)n−1a+b−4+2<(b−1)n+a+b−3a+b−3,

which contradicts δ(G)≥(b−1)n+a+b−3a+b−3.

From what we discussed above, we can draw the conclusion that G′ has a fractional (g − 2, f − 2)-factor. Hence, the proof of Theorem 1 is finished successfully. □

3 Proof of Theorem 2

If G is complete, clearly G has a Hamiltonian fractional (g, f)-factor by the order of graph. We always assume G is not a complete graph below.

Considering the situation of Theorem 2 and the definition of Hamiltonian graph, G admits a Hamiltonian cycle C. Let G′ = G − E(C). Clearly, G includes the desired fractional factor if H includes a fractional (g − 2, f − 2)-factor. Using contradiction, we assume that G′ doesn’t have fractional (g − 2, f − 2)-factor.

Then under the help of Lemma 3, there exists some subset S of V(G′) satisfying

(a−2)|S|+dG′−S(T)−(b−2)|T|≤(f−2)(S)+dG′−S(T)−(g−2)(T)≤−1,(8)

where T = {x ∈ V(G′) − S, dG′-S(x) ≤ b − 3}. Moreover, for each x ∈ T, we get

dG−S(x)≤dG′−S(x)+2≤b−1.(9)

In light of (8) and (9), we infer

b|T|−dG−S(T)>(a−2)|S|.(10)

Since G is not complete, by Lemma 5 we ensure δ(G) ≥ 2t(G)≥2(b−1)+2(b−1)a−2>b. Hence, from (10), we get S ≠ ∅ and |S| ≠ 1. Let λ be the amount of the components of H′ = G[T] which are isomorphic to Kb and let T0 = {x : x ∈ V(H′), dG-S(x) = 0}. Let H be the subgraph gotten from H′ − T0 by deleting these λ components isomorphic to Kb.

If |V(H)| = 0, in view of (10), we obtain

b|T0|+bλ>(a−2)|S|,

and

2≤|S|<b|T0|+bλa−2.

Hence, we infer

|T0|+λ>2(a−2)b.

If ω(G − S) ≥ |T0| + λ > 1, we deduce

t(G)≤|S|ω(G−S)≤|S||T0|+λ<b|T0|+bλ(a−2)(|T0|+λ)=ba−2≤b−1+b−1a−2,

which contradicts to t(G)≥b−1+b−1a−2.

If |T0| + λ = 1, then dG-S(x) + |S| ≥ dG(x) ≥ δ(G) ≥ 2t(G) ≥ 2(b−2)+2(b−1)a−2. We have dG−S(x)≥2(b−2)+2(b−1)a−2−|S|>2(b−2)+2(b−1)a−2−ba−2=2(b−2)+b−2a−2, which contradicts to (9).

Now, we discuss |V(H)| ≥ 1. Let H = H1 ∪ H2, where H1 is the combination of components of H that meets dG-S(x) = b − 1 for each vertex x ∈ V(H1) and H2 = H − H1. By means of Lemma 2, there exists a maximum independent set I1 and the covering set C1 = V(H1) − I1 of H1 such that

|V(H1)|≤(b−1b+1)|I1|(11)

and

|C1|≤(b−1−1b+1)|I1|.(12)

Clearly, δ(H2) ≥ 1 and Δ(H2) ≤ b − 1. Let Tj = {x ∈ V(H2)|dG-S(x) = j} for 1 ≤ j ≤ b − 1. Each component of H2 has a vertex of degree at most b − 2 in G − S from the definitions of H and H2. On the ground of Lemma 1, H2 has a maximal independent set I2 and the covering set C2 = V(H2) − I2 such that

∑j=1b−1(b−j)cj≤∑j=1b−1(b−2)(b−j)ij,(13)

where Cj = |C2∩Tj| and ij = |I2∩Tj| for each j = 1,..., b−1. Set W = V(G) − S − T and U = S ∪ C1 ∪ (NG(I1) ∩ W) ∪ C2 ∪ (NG(I2) ∩ W). We infer

|U|≤|S|+|C1|+∑j=1b−1jij(14)

and

ω(G−U)≥|T0|+λ+|I1|+∑j=1b−1ij.(15)

If ω(G − U) > 1, we yield

|U|≥t(G)ω(G−U).(16)

If ω(G − U) = 1, then (16) also holds.

Using (14), (15) and (16), we obtain

|S|+|C1|+∑j=1b−1jij≥t(G)|T0|+λ+|I1|+∑i=1b−1ij.(17)

In terms of (10), we yield

b|T0|+bλ+|V(H1)|+∑j=1b−1(b−j)ij+∑j=1b−1(b−j)cj>(a−2)|S|.(18)

In light of (17) and (18), we have

|V(H1)|+∑j=1b−1(b−j)cj+(a−2)|C1|>∑j=1b−1((a−2)t(G)−(a−2)j−b+j)ij+((a−2)t(G)−b)(|T0|+λ)+(a−2)t(G)|I1|.(19)

In view of t(G)≥b−1+b−1a−2, we infer (a − 2)t(G) − b ≥ (a − 2)(b − 1) − 1 ≥ 1. Using (19), we deduce

|V(H1)|+∑j=1b−1(b−j)cj+(a−2)|C1|>∑j=1b−1((a−2)t(G)−(a−2)j−b+j)ij+(a−2)t(G)|I1|+1.(20)

According to (11) and (12), we derive

|V(H1)|+(a−2)|C1|≤[(b−1b+1)+(a−2)(b−1−1b+1)]|I1|.

Combining this with (13) and (20), we have

∑j=1b−1(b−2)(b−j)ij+[(b−1b+1)+(a−2)(b−1−1b+1)]|I1|>∑j=1b−1((a−2)t(G)−(a−2)j−b+j)ij+(a−2)t(G)|I1|+1.

Therefore, at least one of the two cases below must be proved to be efficient.

Case 1

(b−1b+1)+(a−2)(b−1−1b+1)>(a−2)t(G)+1. We get

t(G)<b−1−1b+1+b−1b+1a−2−1a−2=b−1+b2−1(b+1)(a−2)+b−a+2(b+1)(a−2)−1a−2<b−1+b2−1(b+1)(a−2)=b−1+b−1a−2,

which contradicts t(G)≥b−1+b−1a−2.

Case 2

There exists at least one j such that

(b−2)(b−j)>(a−2)t(G)−(a−2)j−b+j.

It means

(b−1)(b−j)+(a−2)j>(a−2)t(G).(21)

By j ≤ b − 1 and (21), we yield

b−1+(a−2)(b−1)>(a−2)t(G).

It reveals

t(G)<b−1+b−1a−2.

which is conflictive with t(G)≥b−1+b−1a−2. □

4 Proof of Theorem 3

By means of the hypothesis of Theorem 3, G − X admits a Hamiltonian cycle C for any independent set X in G. Set R = G − X for any independent set X ⊆ V(G). Using the hypothesis of Theorem 3 and the definition of ID- Hamiltonian graph, R contains a Hamiltonian cycle C. Set H = R − E(C). Hence, G admits the desired fractional factor if H has a fractional (g − 2, f − 2)-factor. We assume that H has no fractional (g − 2, f − 2)-factor. Then, there exists some subset S of V(H) = V(R) with

(a−2)|S|+dH−S(T)−(b−2)|T|≤(f−2)(S)+dH−S(T)−(g−2)(T)≤−1,(22)

where T = {x ∈ V(H) − S, dH-S(x) ≤ b − 3}.

If T = ∅, then follow from (22) we obtain

−1≥(a−2)|S|≥0,

which is a contradiction. Thus, T ≠ ∅.

We select x1 ∈ T, so x1 is the vertex with the least degree in R[T]. Set N1 = NR[x1] ∩ T and T1 = T. For i ≥ 2, we continue set Ti = T − ∪1≤j<iNj if T − ∪1≤j<iNj ≠ ∅. In the following, we select xi ∈ Ti such that xi is the vertex with the least degree in R[Ti], and set Ni = NR[xi] ∩ Ti. We continue this procedures until Ti = ∅ for some i, say for i = m + 1. In light of the concept mentioned above, we ensure that {x1, x2,...., xm} is an independent set of R. Obviously, T ≠ ∅ and m ≥ 1.

Let |Ni| = ni, and thus |T|=∑1≤i≤mni. Set U = V(R) − (S ∪ T) and κ(R − S) = t.

Claim 3.1

dR-S(x) ≤ dH-S(x) + 2 ≤ b − 1 for any x ∈ T.

Proof

Using the definition of T and H = R − E(C), we infer

dR−S(x)≤dH−S(x)+2≤(b−3)+2=b−1

for any x ∈ T. □

Claim 3.2

α(G)≤κ(G)−(a+b+1)(a+b−3)4(a−2).

Proof

If α(G)<(a+b+1)(a+b−3)(b+1)2, then by means of (1), we deduce

κ(G)≥(a+b+1)(a+b−3)4(a−2)+(a+b+1)(a+b−3)(b+1)2>(a+b+1)(a+b−3)4(a−2)+α(G).

If α(G)≥(a+b+1)(a+b−3)(b+1)2, then in view of κ(G) ≥ (b+1)2+4(a−2)4(a−2)α(G), we get

κ(G)≥(b+1)2+4(a−2)4(a−2)α(G)=(b+1)24(a−2)α(G)+α(G)≥(b+1)24(a−2)(a+b+1)(a+b−3)(b+1)2+α(G)=(a+b+1)(a+b−3)4(a−2)+α(G).

Now, Claim 3.2 is completed. □

Claim 3.3

U ≠ ∅ or m ≠ 1.

Proof

Assume that m = 1 and U = ∅. In terms of (22), Claim 3.1 and the selection rule of x1, we yield

−1≥(a−2)|S|+dH−S(T)−(b−2)|T|≥(a−2)|S|+dR−S(T)−2|T|−(b−2)|T|=(a−2)|S|+dR−S(T)−b|T|=(a−2)|S|+n1(n1−1)−bn1,

which means

|S|≤−n12+(b+1)n1−1a−2.(23)

In view of (23), Claim 3.2, R = G − X and |X| ≤ α(G), we have

|V(G)|=|V(R)|+|X|=|S|+|T|+|X|=|S|+n1+|X|≤−n12+(b+1)n1−1a−2+n1+α=−n12+(a+b−1)n1−1a−2+α≤(a+b−1)2−44(a−2)+α=(a+b−1)(a+b−3)4(a−2)+α≤κ(G),

which contradicts to |V(G)| > κ(G). Claim 3.3 is hold. □

Claim 3.4

dR−S(T)≥∑1≤i≤mni(ni−1)+mt2.

Proof

The selection rule of xi reveals that all vertices in Ni have degree at least ni − 1 in R[Ti]. Therefore, we obtain

∑1≤i≤m(∑x∈NidR[Ti](x))≥∑1≤i≤mni(ni−1).(24)

On the left of (24), an edge joining x ∈ Ni and y ∈ Nj (i < j) is counted only once, i.e., it is counted in the term dR[Ti](x) but not in dR[Tj](y). Therefore, we derive

dR−S(T)≥∑1≤i≤mni(ni−1)+∑1≤i<j≤meR(Ni,Nj)+eR(T,U).(25)

In light of κ(R − S) = t and Claim 3.3, we yield

eR(Ni,∪j≠iNj)+eR(Ni,U)≥t(26)

for any Ni(i = 1, 2,..., m). By means of (26), we get

∑1≤i≤m(eR(Ni,∪j≠iNj)+eR(Ni,U))=2(∑1≤i<j≤meR(Ni,Nj)+eR(T,U))≥mt,

which implies

∑1≤i<j≤meR(Ni,Nj)+eR(T,U)≥mt2.(27)

Using (25) and (27), we infer

dR−S(T)≥∑1≤i≤mni(ni−1)+mt2.

Therefore, Claim 3.4 is proved. □

In terms of |T|=∑1≤i≤mni, (22), Claim 3.1 and Claim 3.4, we have

−1≥(a−2)|S|+dH−S(T)−(b−2)|T|≥(a−2)|S|+dR−S(T)−2|T|−(b−2)|T|=(a−2)|S|+dR−S(T)−b|T|≥(a−2)|S|+∑1≤i≤mni(ni−1)+mt2−b∑1≤i≤mni=(a−2)|S|+∑1≤i≤m(ni2−(b+1)ni)+mt2,
i.e.,
−1≥(a−2)|S|+∑1≤i≤m(ni2−(b+1)ni)+mt2.(28)

Let f(ni)=ni2−(b+1)ni. It is easy to say that min{f(ni)}=−(b+1)24. Hence, we infer

ni2−(b+1)ni≥−(b+1)24.(29)

According to (28) and (29), we deduce

−1≥(a−2)|S|+∑1≤i≤m(−(b+1)24)+mt2=(a−2)|S|−(b+1)2m4+mt2.(30)

Claim 3.5

−(b+1)24+t2<0.

Proof

Suppose that −(b+1)24+t2≥0. In view of (30), m ≥ 1 and |S|≥ 0, we get

−1≥(a−2)|S|−(b+1)2m4+mt2≥0,

a contradiction. Then proof of Claim 3.5 is completed. □

Note that {x1, x2,..., xm} is an independent set of R. We obtain

α(R[T])≥m.(31)

In view of R = G − X and (32), we yield

α(G)≥α(R)≥α(R[T])≥m.(32)

Note that R = G − X and κ(R − S) = t. It is obvious that

κ(G)≤κ(G−X)+|X|=κ(R)+|X|≤κ(R−S)+|S|+|X|=t+|S|+|X|.(33)

In light of (30), (32), (33), |X| ≤ α(G), Claim 3.5, κ(G) ≥ (b+1)2+4(a−2)2,andκ(G)≥(b+1)2+4(a−2)4(a−2)α(G), we have

−1≥(a−2)|S|−(b+1)2m4+mt2=(a−2)|S|+(−(b+1)24+t2)m≥(a−2)(κ(G)−|X|−t)+(−(b+1)24+t2)α(G)≥(a−2)(κ(G)−α(G)−t)+(−(b+1)24+t2)α(G)≥(a−2)(κ(G)−4(a−2)κ(G)(b+1)2+4(a−2)−t)+(−(b+1)24+t2)4(a−2)κ(G)(b+1)2+4(a−2)=(a−2)t(2κ(G)(b+1)2+4(a−2)−1)≥0,

a contradiction. Therefore, we finish proving Theorem 3. □

  1. Conflict of interest

    Conflict of Interests: It is declared that there is no conflict of interests regarding the publication of this paper.

Acknowledgement

The constructive comments from the reviewers to improve the paper are sincerely appreciated.

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Received: 2017-1-26
Accepted: 2017-3-13
Published Online: 2017-8-3

© 2017 W. Gao et al.

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

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https://doi.org/10.1515/phys-2017-0057
Keywords for this article
software definition network; data transmission; fractional factor; toughness; Hamiltonian fractional factors
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  81. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  82. A mathematical/physics model to measure the role of information and communication technology in some economies: the Chinese case
  83. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  84. Numerical modeling of the thermoelectric cooler with a complementary equation for heat circulation in air gaps
  85. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  86. On the libration collinear points in the restricted three – body problem
  87. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  88. Research on Critical Nodes Algorithm in Social Complex Networks
  89. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  91. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  93. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  95. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  97. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  98. A privacy-preserving parallel and homomorphic encryption scheme
  99. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  100. Random walk-based similarity measure method for patterns in complex object
  101. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  102. A Mathematical Study of Accessibility and Cohesion Degree in a High-Speed Rail Station Connected to an Urban Bus Transport Network
  103. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  104. Design and Simulation of the Integrated Navigation System based on Extended Kalman Filter
  105. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  106. Oil exploration oriented multi-sensor image fusion algorithm
  107. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  108. Analysis of Product Distribution Strategy in Digital Publishing Industry Based on Game-Theory
  109. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  110. Expanded Study on the accumulation effect of tourism under the constraint of structure
  111. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  112. Unstructured P2P Network Load Balance Strategy Based on Multilevel Partitioning of Hypergraph
  113. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  114. Research on the method of information system risk state estimation based on clustering particle filter
  115. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  116. Demand forecasting and information platform in tourism
  117. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  118. Physical-chemical properties studying of molecular structures via topological index calculating
  119. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  120. Local kernel nonparametric discriminant analysis for adaptive extraction of complex structures
  121. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  122. City traffic flow breakdown prediction based on fuzzy rough set
  123. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  124. Conservation laws for a strongly damped wave equation
  125. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  126. Blending type approximation by Stancu-Kantorovich operators based on Pólya-Eggenberger distribution
  127. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  128. Computing the Ediz eccentric connectivity index of discrete dynamic structures
  129. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  130. A discrete epidemic model for bovine Babesiosis disease and tick populations
  131. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  132. Study on maintaining formations during satellite formation flying based on SDRE and LQR
  133. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  134. Relationship between solitary pulmonary nodule lung cancer and CT image features based on gradual clustering
  135. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  137. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  141. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  142. After notes on self-similarity exponent for fractal structures
  143. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  144. Excitation probability and effective temperature in the stationary regime of conductivity for Coulomb Glasses
  145. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  146. Comparisons of feature extraction algorithm based on unmanned aerial vehicle image
  147. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  148. Research on identification method of heavy vehicle rollover based on hidden Markov model
  149. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  150. Classifying BCI signals from novice users with extreme learning machine
  151. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  152. Topics on data transmission problem in software definition network
  153. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  154. Statistical inferences with jointly type-II censored samples from two Pareto distributions
  155. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  156. Estimation for coefficient of variation of an extension of the exponential distribution under type-II censoring scheme
  157. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  158. Analysis on trust influencing factors and trust model from multiple perspectives of online Auction
  159. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  160. Coupling of two-phase flow in fractured-vuggy reservoir with filling medium
  161. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  162. Production decline type curves analysis of a finite conductivity fractured well in coalbed methane reservoirs
  163. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
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  165. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
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  167. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  168. Research on carbonate reservoir interwell connectivity based on a modified diffusivity filter model
  169. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  170. The method of the spatial locating of macroscopic throats based-on the inversion of dynamic interwell connectivity
  171. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  172. Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method
  173. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  174. A volumetric ablation model of EPDM considering complex physicochemical process in porous structure of char layer
  175. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  176. Numerical simulation on ferrofluid flow in fractured porous media based on discrete-fracture model
  177. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  178. Macroscopic lattice Boltzmann model for heat and moisture transfer process with phase transformation in unsaturated porous media during freezing process
  179. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  180. Modelling of intermittent microwave convective drying: parameter sensitivity
  181. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
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  183. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  184. Simulation of counter-current imbibition in water-wet fractured reservoirs based on discrete-fracture model
  185. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  186. Investigation effect of wettability and heterogeneity in water flooding and on microscopic residual oil distribution in tight sandstone cores with NMR technique
  187. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  188. Analytical modeling of coupled flow and geomechanics for vertical fractured well in tight gas reservoirs
  189. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  190. Special Issue: Ever New "Loopholes" in Bell’s Argument and Experimental Tests
  191. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  192. The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  193. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  194. Erratum to: The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  195. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  196. Rhetoric, logic, and experiment in the quantum nonlocality debate
  197. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  198. What If Quantum Theory Violates All Mathematics?
  199. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  200. Relativity, anomalies and objectivity loophole in recent tests of local realism
  201. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
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  203. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  204. Bohr against Bell: complementarity versus nonlocality
  205. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  206. Is Einsteinian no-signalling violated in Bell tests?
  207. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  208. Bell’s “Theorem”: loopholes vs. conceptual flaws
  209. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  210. Nonrecurrence and Bell-like inequalities
  211. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  212. Three-dimensional computer models of electrospinning systems
  213. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  214. Electric field computation and measurements in the electroporation of inhomogeneous samples
  215. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  216. Modelling of magnetostriction of transformer magnetic core for vibration analysis
  217. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  218. Comparison of the fractional power motor with cores made of various magnetic materials
  219. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  220. Dynamics of the line-start reluctance motor with rotor made of SMC material
  221. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  222. Inhomogeneous dielectrics: conformal mapping and finite-element models
  223. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  224. Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
  225. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  226. Detection of inter-turn short-circuit at start-up of induction machine based on torque analysis
  227. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  228. Current superimposition variable flux reluctance motor with 8 salient poles
  229. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  230. Modelling axial vibration in windings of power transformers
  231. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  232. Field analysis & eddy current losses calculation in five-phase tubular actuator
  233. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  234. Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
  235. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  236. Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
  237. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  238. Field-circuit analysis and measurements of a single-phase self-excited induction generator
  239. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  240. A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
  241. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  242. Field-based optimal-design of an electric motor: a new sensitivity formulation
  243. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  244. Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
  245. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  246. Virtual reality as a new trend in mechanical and electrical engineering education
  247. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  248. Holonomicity analysis of electromechanical systems
  249. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  250. An accurate reactive power control study in virtual flux droop control
  251. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  252. Localized probability of improvement for kriging based multi-objective optimization
  253. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  254. Research of influence of open-winding faults on properties of brushless permanent magnets motor
  255. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  256. Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
  257. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  258. Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
  259. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  260. Detection of inter-turn faults in transformer winding using the capacitor discharge method
  261. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  262. A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
  263. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  264. Lamination effects on a 3D model of the magnetic core of power transformers
  265. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  266. Detection of vertical disparity in three-dimensional visualizations
  267. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  268. Calculations of magnetic field in dynamo sheets taking into account their texture
  269. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  270. 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
  271. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  272. Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
  273. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  274. Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
  275. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  276. Speed control of an induction motor by 6-switched 3-level inverter
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