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Neighborhood condition for all fractional (g, f, n′, m)-critical deleted graphs

Wei Gao
/ Yunqing Zhang
/ Yaojun Chen
Published Online: 2018-08-20 | DOI: https://doi.org/10.1515/phys-2018-0071

Abstract

In data transmission networks, the availability of data transmission is equivalent to the existence of the fractional factor of the corresponding graph which is generated by the network. Research on the existence of fractional factors under specific network structures can help scientists design and construct networks with high data transmission rates. A graph G is named as an all fractional (g, f, n′, m)-critical deleted graph if the remaining subgraph keeps being an all fractional (g, f, m)-critical graph, despite experiencing the removal of arbitrary n′ vertices of G. In this paper, we study the relationship between neighborhood conditions and a graph to be all fractional (g, f, n′, m)-critical deleted. Two sufficient neighborhood conditions are determined, and furthermore we show that the conditions stated in the main results are sharp.

PACS: 02.10.Ox; 07.05.Mh

1 Introduction

The classic fractional factor problem is usually regarded as an extension of illustrious cardinality matching which is one of the hot topics in graph theory and operations research. Its extensive applications can be found in many domains like combinatorial polyhedron, scheduling and designing of network. For instance, we send some large data packets to several destinations through channels of the data transmission network, and the efficiency will be improved if the large data packets can be categorized into more smaller ones. The problem of feasibility allocations of data packets can be considered as the existence of fractional flow in the network, and it can be transformed to the fractional factor problem in the network graph.

In particular, the whole network can be modelled as a graph in which each vertex represents a site and each edge denotes a channel. In normal networks, the route of data transmission is picked out based on the shortest path between vertices. A few advances of data transmission in networks have been manifested in recent years. Rolim et al. [1] improved the data transmission by studying an urban sensing problem in view of opportunistic networks. Vahidi et al. [2] considered unmanned aerial vehicles and proposed the high-mobility airborne hyperspectral data transmission algorithm. Miridakis et al. [3] adopted a cost-effective solution to study the dual-hop cognitive secondary relaying system. Streaming data transmission on a discrete memoryless channel was discussed by Lee et al. [4]. But, in the context of software definition network (SDN), the path between vertices in data transmission relies on the current network flow computation. With minimum transmission congestion in the current moment, the transmission route is chosen. In this view, the framework of data transmission problem in SDN equals to the existence of the fractional factor avoiding certain subgraphs.

Throughout this paper, we only consider the simple graph which corresponds to a data transmission network. Let G = (V(G), then E(G)) be a graph with vertex set V(G) and edge set E(G). For any xV(G), we denote dG(x) and NG(x) by the degree and the open neighborhood of x in G, separately. Set NG[x] = NG(x) ∪ {x} as the closed neighborhood of x in G. We denote by G[S] the subgraph of G induced by S, and GS = G[V(G) ∖ S] for any SV(G). For two vertex-disjoint subsets S, TV(G), we set eG(S, T) = |{e = xy|xS, yT}|. We denote the minimum degree of G by δ(G) = min{dG(x) : xV(G)}. As a simple expression, we take d(x) to express dG(x) for xV(G). More terminologies and notations used but undefined in our article can be found in book Bondy and Mutry [5].

Let g and f be integer-valued functions on V(G) satisfying 0 ≤ g(x) ≤ f(x) for any xV(G). A fractional (g, f)-factor can be considered as a function h that gives to every edge of a graph G a real number in [0, 1] with $\begin{array}{}g\left(x\right)\le \sum _{e\in E\left(x\right)}h\left(e\right)\le f\left(x\right)\end{array}$ for each vertex x. If g(x) = f(x) for any xV(G), a fractional (g, f)-factor is a fractional f-factor. If g(x) = a and f(x) = b for any xV(G), then a fractional (g, f)-factor is a fractional [a, b]-factor. Furthermore, a fractional (g, f)-factor becomes a fractional k-factor if g(x) = f(x) = k (where k ≥ 1 is an integer) for any xV(G). Then, we always assume that n is the order of G, namely, n = |V(G)|.

If G has a fractional p-factor for every p : V(G) → ℕ such that g(x) ≤ p(x) ≤ f(x) for any xV(G), G possesses all fractional (g, f)-factors. If g(x) = a, f(x) = b for each xV(G) and G possesses all fractional (g, f)-factors, then G has all fractional [a, b]-factors.

Lu [6] determined the sufficient and necessary condition for a graph that has all fractional (g, f)-factors.

Theorem 1

(Lu [6]) Assume G to be a graph, g, f : V(G) → ℤ+ be integer functions so that g(x) ≤ f(x) for all xV(G), so G has all fractional (g, f)-factors. If and only if for any subset SV(G), we have

$g(S)−f(T)+∑x∈TdG−S(x)≥0,$

where T = {xV(G) – S|dGS(x) < f(x)}.

Obviously, this sufficient and necessary condition equals to the following version.

Theorem 2

Assume G be a graph, g, f : V(G)→ ℤ+ be integer functions so that g(x) ≤ f(x) for all xV(G). As a result, G has all fractional (g, f)-factors. If and only if

$g(S)−f(T)+∑x∈TdG−S(x)≥0$

for arbitrary non-disjoint subsets S, TV(G).

Set g(x) = a, f(x) = b for each xV(G), we immediately get the below corollary from Theorem 1.

Corollary 1

Let G be a graph and ab be two positive integers. Then G has all fractional [a, b]-factors if and only if for any subset SV(G), we have

$a|S|−b|T|+∑x∈TdG−S(x)≥0,$

where T = {xV(G) – S|dGS(x) < b}.

Also, this sufficient and necessary condition in Corollary 1 for all fractional [a, b]-factors equals to the following version.

Corollary 2

Let G be a graph and ab be two positive integers. Then G has all fractional [a, b]-factors if and only if

$a|S|−b|T|+∑x∈TdG−S(x)≥0$

for arbitrary non-disjoint subsets S, TV(G).

Zhou and Zhang [7] presented the sufficient and necessary situation for a graph with all fractional (g, f)-factors excluding a subgraph H.

Theorem 3

(Zhou and Zhang [7]) Let G be a graph and H be a subgraph of G. Let g, f : V(G) → ℤ+ be two integer-valued functions with g(x) ≤ f(x) for each xV(G). Then G admits all fractional (g, f)-factors excluding H if and only if

$g(S)−f(T)+∑x∈TdG−S(x)≥∑x∈TdH(x)−eH(S,T)$

for arbitrary subset S of V(G), where T = {xV(G) – S|dGS(x) – dH(x) + eH(x, S) < f(x)}.

Clearly, this sufficient and necessary condition has the following equal version.

Theorem 4

Let G be a graph and g, f : V(G) → ℤ+ be two integer-valued functions with g(x) ≤ f(x) for each xV(G). Let H be a subgraph of G. Then G admits all fractional (g, f)-factors excluding H if and only if

$g(S)−f(T)+∑x∈TdG−S(x)≥∑x∈TdH(x)−eH(S,T)$

for arbitrary non-disjoint subsets S, TV(G).

Set g(x) = a, f(x) = b for each xV(G), we immediate get the Corollary below.

Corollary 3

Let G be a graph, and a and b be two integers where 1 ≤ ab. Let H be a subgraph of G. Then G admits all fractional [a, b]-factors excluding H if and only if

$a|S|+∑x∈T(dG−S(x)−dH(x)+eH(x,S)−b)≥0,$

where T = {xV(G) – S|dGS(x) – dH(x) + eH(x, S) < b}.

Again, this sufficient and necessary condition can be stated as follows.

Corollary 4

Let a and b be two integers with 1 ≤ ab, and let G be a graph. Let H be a subgraph of G. So G admits all fractional [a, b]-factors excluding H if and only if

$a|S|+∑x∈T(dG−S(x)−dH(x)+eH(x,S)−b)≥0$

for arbitrary non-disjoint subsets S, TV(G).

Zhou and Sun [8] introduced the concept of all fractional (a, b, n′)-critical graphs. A graph G is named as an all fractional (a, b, n′)-critical graph if the remaining graph of G has all fractional [a, b]-factors, even though any n′ vertices of G are deleted. Also, they presented the necessary and valuable situation for a graph to be all fractional (a, b, n′)-critical.

Theorem 5

(Zhou and Sun [8]) Let a, b and n′ be nonnegative integers with 1 ≤ ab, and let G be a graph of order n with na + n′ + 1. Then G is all fractional (a, b, n′)-critical if and only if for any SV(G) with |S| ≥ n′,

$a|S|+∑x∈TdG−S(x)−b|T|≥an′,$

where T = {x : xV(G) ∖ S, dGS(x) < b}.

Equally, the above necessary and sufficient condition can be re-written as follows.

Theorem 6

Assume a, b and n′ be nonnegative integers with 1 ≤ ab, hence suppose G is a graph of order n with na + n′ + 1. Then, G is all fractional (a, b, n′)-critical if and only if

$a|S|+∑x∈TdG−S(x)−b|T|≥an′$

for arbitrary non-disjoint subsets S, TV(G) with |S| ≥ n′.

More results from this topic, regarding fractional factor, fractional deleted graphs, fractional critical and other network applications can be found in Zhou et al. [9], Jin [10], Gao and Wang [11] and [12], Gao et al. [13], [14] and [15], and Guirao and Luo [16].

In this paper, we first introduce some new concepts. A graph G is named as an all fractional (g, f, m)-deleted graph if the remaining graph of G has all fractional (g, f)-factors, when any m edges of G are deleted. If for any xV(G), we have g(x) = a and f(x) = b, then an all fractional (g, f, m)-deleted graph becomes an all fractional (a, b, m)-deleted graph, i.e., a graph G is named as an all fractional (a, b, m)-deleted graph if the remaining graph of G has all fractional [a, b]-factors, when any m edges of G are deleted. Following this definition, we clearly see that Theorem 3 and Theorem 4 show the necessary and sufficient condition of all fractional (g, f, m)-deleted graph with |E(H)| = m respectively, and Corollary 3 and Corollary 4 present the necessary and sufficient condition of all fractional (a, b, m)-deleted graph respectively.

Next, we combine two concepts, all fractional (g, f, m)-deleted graph and all fractional (g, f, m)-critical graph together. A graph G is named as an all fractional (g, f, n′, m)-critical deleted graph if the remaining graph of G is still an all fractional (g, f, m)-deleted graph, when any n′ vertices of G are deleted. If g(x) = a, f(x) = b for each xV(G), every fractional (g, f, n′, m)-critical deleted graph becomes all fractional (a, b, n′, m)-critical deleted graph, it means, after deleting any n′ vertices of G the remaining graph of G is still an all fractional (a, b, m)-deleted graph.

The concept of all fractional (g, f, n′, m)-critical deleted graph reflects the feasibility of data transmission in data transmission networks, in the case of some sites or channels being damaged. At the same time, it also provides theoretical support for SDN: in the peak data transmission, some sites and communications are often in the situation of information congestion, so we regard the site and channel in a blocked state as the vertices and edges that need to be deleted. Then, we consider the existence of fractional factors in the resulting subgraph. Thus, it inspires us to study the sufficient condition of all fractional (g, f, n′, m)-critical deleted graph from the graph structure prospect. It will imply which structures of network can ensure the success of data transmission, and the theoretical results obtained in our article can help us carry out the network design.

In this paper, we explore relations between the neighborhood union condition and all fractional (g, f, n′, m)-critical deleted graphs. The first key result can be formulated as follows.

Theorem 7

Assume a, b, n′ and m are four non-negative integers satisfying 2 ≤ ab. Assume G to be a graph with $\begin{array}{}\delta \left(G\right)\ge \frac{\left(a+b-1{\right)}^{2}+4b}{4a}+m+{n}^{\prime }\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}n\ge \frac{2\left(a+b\right)\left(a+b+m-1\right)}{a}+{n}^{\prime }.\end{array}$ Let g and f be two integer functions which are defined on V(G) such that ag(x) ≤ f(x) ≤ b for each xV(G). If |NG(u) ∪ NG(v)| ≥ $\begin{array}{}\frac{bn+a{n}^{\prime }}{a+b}\end{array}$ for arbitrary two nonadjacent vertices u and v in G, then G is all fractional (g, f, n′, m)-critical deleted.

Set n′ = 0 in Theorem 7, then we get the following corollary.

Corollary 5

Assume a, b and m be three non-negative integers satisfying 2 ≤ ab. Assume G is a graph with $\begin{array}{}\delta \left(G\right)\ge \frac{\left(a+b-1{\right)}^{2}+4b}{4a}+m\end{array}$ and $\begin{array}{}n\ge \frac{2\left(a+b\right)\left(a+b+m-1\right)}{a}.\end{array}$ Let g and f be two integer functions whose definition can be found on V(G) such that ag(x) ≤ f(x) ≤ b for each xV(G). If |NG(u) ∪ NG(v)| ≥ $\begin{array}{}\frac{bn}{a+b}\end{array}$ for arbitrary two nonadjacent vertices u and v in G, then G is all fractional (g, f, m)-deleted.

By setting m = 0 in Theorem 7, the following corollary is obtained.

Corollary 6

Assume a, b and n′ to be three non-negative integers satisfying 2 ≤ ab. Let G be a graph with δ(G) ≥ $\begin{array}{}\frac{\left(\left(a+b-1{\right)}^{2}+4b\right)}{4a}+{n}^{\prime }\end{array}$ and $\begin{array}{}n\ge \frac{2\left(a+b\right)\left(a+b-1\right)}{a}+{n}^{\prime }.\end{array}$ Let g and f be two integer functions whose definition can be found on V(G) such that ag(x) ≤ f(x) ≤ b for each xV(G). If |NG(u) ∪ NG(v)| ≥ $\begin{array}{}\frac{bn+a{n}^{\prime }}{a+b}\end{array}$ for arbitrary two nonadjacent vertices u and v in G, then G is all fractional (g, f, n′)-critical.

Set g(x) = a and f(x) = b for any xV(G), then we get the following condition for all fractional (a, b, n′, m)-critical deleted graph.

Corollary 7

Let a, b, n′ and m be four non-negative integers satisfying 2 ≤ ab. Let G be a graph with $\begin{array}{}\delta \left(G\right)\ge \frac{\left(a+b-1{\right)}^{2}+4b}{4a}\end{array}$ + m + n′ and $\begin{array}{}n\ge \frac{2\left(a+b\right)\left(a+b+m-1\right)}{a}+{n}^{\prime }.\end{array}$ If |NG(u) ∪ NG(v)| ≥ $\begin{array}{}\frac{bn+a{n}^{\prime }}{a+b}\end{array}$ for arbitrary two nonadjacent vertices u and v in G, then G is all fractional (a, b, n′, m)-critical deleted.

Set n′ = 0 in Corollary 7, then the corollary below can be obtained.

Corollary 8

Let a, b and m be three non-negative integers satisfy 2 ≤ ab. Let G be a graph with $\begin{array}{}\delta \left(G\right)\ge \frac{\left(a+b-1{\right)}^{2}+4b}{4a}\end{array}$ + m and n$\begin{array}{}\frac{2\left(a+b\right)\left(a+b+m-1\right)}{a}\end{array}$. If |NG(u) ∪ NG(v)| ≥ $\begin{array}{}\frac{bn}{a+b}\end{array}$ for arbitrary two nonadjacent vertices u and v in G, then G is all fractional (a, b, m)-deleted.

Set m = 0 in Corollary 7, then the following corollary is obtained.

Corollary 9

Let a, b and n′ be three non-negative integers satisfy 2 ≤ ab. Let G be a graph with δ(G) ≥ $\begin{array}{}\frac{\left(a+b-1{\right)}^{2}+4b}{4a}\end{array}$ + n′ and $\begin{array}{}n\ge \frac{2\left(a+b\right)\left(a+b-1\right)}{a}+{n}^{\prime }.\end{array}$ If |NG(u) ∪ NG(v)| ≥ $\begin{array}{}\frac{bn+a{n}^{\prime }}{a+b}\end{array}$ for arbitrary two nonadjacent vertices u and v in G, then G is all fractional (a, b, n′)-critical.

Our second main result is stated as follows.

Theorem 8

Let a, b, n′ and m be four non-negative integers satisfying 2 ≤ ab. Let G be a graph with order $\begin{array}{}n\ge \frac{\left(a+2b-3\right)\left(a+b-2\right)}{a}+\frac{a{n}^{\prime }}{a-1}+m.\end{array}$ Let g and f be two integer functions whose definition can be found on V(G) such that ag(x) ≤ f(x) ≤ b for each xV(G). Suppose that

$NG(X)=V(G)if|X|≥⌊(an−a−an′−2m)n(a+b−1)(n−1)⌋;$

or

$NG(X)≥(a+b−1)(n−1)an−a−an′−2m|X|if|X|<⌊(an−a−an′−2m)n(a+b−1)(n−1)⌋$

for any subset XV(G). Then G is all fractional (g, f, n′, m)-critical deleted.

Set n′ = 0 in Theorem 8, then we yield the following corollary.

Corollary 10

Let a, b, and m be three non-negative integers satisfy 2≤ ab. Let G be a graph with order n$\begin{array}{}\frac{\left(a+2b-3\right)\left(a+b-2\right)}{a}\end{array}$ + m. Let g and f be two integer functions defined on V(G) such that ag(x) ≤ f(x) ≤ b for each xV(G). Suppose that

$NG(X)=V(G)if|X|≥⌊(an−a−2m)n(a+b−1)(n−1)⌋;$

or

$NG(X)≥(a+b−1)(n−1)an−a−2m|X|if|X|<⌊(an−a−2m)n(a+b−1)(n−1)⌋$

for any subset XV(G). Then G is all fractional (g, f, m)-deleted.

Set m = 0 in Theorem 8, then the Corollary below can be inferred.

Corollary 11

Let a, b and n′ be three non-negative integers satisfy 2≤ ab. Let G be a graph with order n$\begin{array}{}\frac{\left(a+2b-3\right)\left(a+b-2\right)}{a}+\frac{a{n}^{\prime }}{a-1}.\end{array}$ Let g and f be two integer functions defined on V(G) such that ag(x) ≤ f(x) ≤ b for each xV(G). Suppose that

$NG(X)=V(G)if|X|≥⌊(an−a−an′)n(a+b−1)(n−1)⌋;$

or

$NG(X)≥(a+b−1)(n−1)an−a−an′|X|if|X|<⌊(an−a−an′)n(a+b−1)(n−1)⌋$

for any subset XV(G). Then G is all fractional (g, f, n′)-critical.

Set g(x) = a and f(x) = b for any xV(G), then we get the following condition for an all fractional (a, b, n′, m)-critical deleted graph.

Corollary 12

Let a, b, n′ and m be four non-negative integers satisfying 2 ≤ ab. Let G be a graph with order $\begin{array}{}n\ge \frac{\left(a+2b-3\right)\left(a+b-2\right)}{a}+\frac{a{n}^{\prime }}{a-1}+m.\end{array}$ Suppose that

$NG(X)=V(G)if|X|≥⌊(an−a−an′−2m)n(a+b−1)(n−1)⌋;$

or

$NG(X)≥(a+b−1)(n−1)an−a−an′−2m|X|if|X|<⌊(an−a−an′−2m)n(a+b−1)(n−1)⌋$

for any subset XV(G). Then G is all fractional (a, b, n′, m)-critical deleted.

Set n′ = 0 in Corollary 12, then we have the following corollary.

Corollary 13

Let a, b and m be three non-negative integers satisfy 2 ≤ ab. Let G be a graph with order n$\begin{array}{}\frac{\left(a+2b-3\right)\left(a+b-2\right)}{a}\end{array}$ + m. Suppose that

$NG(X)=V(G)if|X|≥⌊(an−a−2m)n(a+b−1)(n−1)⌋;$

or

$NG(X)≥(a+b−1)(n−1)an−a−2m|X|if|X|<⌊(an−a−2m)n(a+b−1)(n−1)⌋$

for any subset XV(G). Then G is all fractional (a, b, m)-deleted.

Set m = 0 in Corollary 12, then we deduce the following corollary.

Corollary 14

Let a, b and n′ be three non-negative integers satisfying 2 ≤ ab. Let G be a graph with order n$\begin{array}{}\frac{\left(a+2b-3\right)\left(a+b-2\right)}{a}+\frac{a{n}^{\prime }}{a-1}.\end{array}$ Suppose that

$NG(X)=V(G)if|X|≥⌊(an−a−an′)n(a+b−1)(n−1)⌋;$

or

$NG(X)≥(a+b−1)(n−1)an−a−an′|X|if|X|<⌊(an−a−an′)n(a+b−1)(n−1)⌋$

for any subset XV(G). Then G is all fractional (a, b, n′)-critical deleted.

2.1 The necessary and sufficient condition of all fractional (g, f, n′, m)-critical deleted graphs

To prove the main results in our paper, we first determine the necessary and sufficient condition of all fractional (g, f, n′, m)-critical deleted graphs which is manifested as follows.

Theorem 9

a, b, m and n′ are assumed to be nonnegative integers satisfying 1 ≤ ab, and G is let to be a graph of order n with nb + n′ + m + 1. Let g, f : V(G) → ℤ+ be two valued functions with ag(x) ≤ f(x) ≤ b for each xV(G), and H will be a subgraph of G with m edges. After that, G is all fractional (g, f, n′, m)-critical deleted if and only if for any SV(G) with |S| ≥ n′,

$g(S)−f(T)+∑x∈TdG−S(x)≥maxU⊆S,|U|=n′,H⊆E(G−U),|H|=m{g(U)+∑x∈TdH(x)−eH(S,T)},$

where

$T={x:x∈V(G)∖S,dG−S(x)−dH(x)+eH(x,S)(1)

The equal version of Theorem 9 is stated as follows.

Theorem 10

a, b, m and n′ are assumed to be nonnegative integers satisfying 1 ≤ ab, and let G be a graph of order n with nb + n′ + m + 1. Let g, f : V(G) → ℤ+ be two valued functions with ag(x) ≤ f(x) ≤ b for each xV(G), and H be a subgraph of G with m edges. Then, G is all fractional (g, f, n′, m)-critical deleted if and only if

$g(S)−f(T)+∑x∈TdG−S(x)≥maxU⊆S,|U|=n′,H⊆E(G−U),|H|=m{g(U)+∑x∈TdH(x)−eH(S,T)},$

for any non-disjoint subsets S, TV(G) with |S| ≥ n′.

Set g(x) = a and f(x) = b for every xV(G), and the following corollary on the necessary and sufficient condition of all fractional (a, b, n′, m)-critical deleted graph is obtained.

Corollary 15

Assume a, b, m and n′ be nonnegative integers satisfying 1 ≤ ab, and G be a graph of order n with na + n′ + m + 1. H is assumed to be a subgraph of G with m edges. G is all fractional (a, b, n′, m)-critical deleted if and only if for any SV(G) with |S| ≥ n′,

$a|S|+∑x∈T(dG−S(x)−dH(x)+eH(x,S)−b)≥an′,$

where T = {xV(G) – S|dGS(x) – dH(x) + eH(x, S) < b}.

Clearly, the above necessary and sufficient condition has the equal version which is stated as follows.

Corollary 16

Assume a, b, m and n′ to be nonnegative integers satisfying 1 ≤ ab, and G a graph of order n with na + n′ + m + 1. H is assumed to be a subgraph of G with m edges. G is all fractional (a, b, n′, m)-critical deleted if and only if

$a|S|+∑x∈T(dG−S(x)−dH(x)+eH(x,S)−b)≥an′$

for any non-disjoint subsets S, TV(G) with |S| ≥ n′.

Now, we present the detailed proof of Theorem 9.

Proof of Theorem 9

Assume USV(G) with |U| = n′. Let S′ = SU and G′ = GU. Then we have G′ – S′ = GS. Moreover, let

$T′={x:x∈V(G′)∖S′,dG′−S′(x)−dH(x) +eH(x,S′)(2)

then by the definition of (1), we get T′ = T. Furthermore, we infer

$g(S′)+∑x∈T′(dG′−S′(x)−dH(x)+eH(x,S′)−f(x))=g(S)+∑x∈T(dG−S(x)−dH(x)+eH(x,S)−f(x))−g(U),$

so that

$g(S′)+∑x∈T′(dG′−S′(x)−dH(x)+eH(x,S′)−f(x))≥0⇔g(S)−f(T)+∑x∈TdG−S(x)≥maxU⊆S,|U|=n′,H⊆E(G−U),|H|=m{g(U)+∑x∈TdH(x)−eH(S,T)}.$(3)

First, assume G be all fractional (g, f, n′, m)-critical deleted, and let S, S′, U, T′ and G′ be described as the above. So G′ = GU is all fractional (g, f, m)-deleted, thus in terms of Theorem 3, we have

$g(S′)+∑x∈T′(dG′−S′(x)−dH(x)+eH(x,S′)−f(x))≥0.$

Hence, using (3), we infer that

$g(S)−f(T)+∑x∈TdG−S(x)≥maxU⊆S,|U|=n′,H⊆E(G−U),|H|=m{g(U)+∑x∈TdH(x)−eH(S,T)}$

holds for any SV(G) with |S| ≥ n′.

Conversely, suppose that

$g(S)−f(T)+∑x∈TdG−S(x)≥maxU⊆S,|U|=n′,H⊆E(G−U),|H|=m{g(U)+∑x∈TdH(x)−eH(S,T)}$

holds for any SV(G) with |S| ≥ n′. Let UV(G) satisfy |U| = n′ and G′ = GU. For any S′ ⊆ V(G′), define T′ according to (2) and let S = S′ ∪ U. Thus, S′ = SU, and in light of (3) we have

$g(S′)+∑x∈T′(dG′−S′(x)−dH(x)+eH(x,S′)−f(x))≥0$

holds for any S′ ⊆ V(G′). It implies that G′ = GU is an all fractional (g, f, m)-deleted graph by Theorem 3. And, this establishes for any UV(G) with |U| = n′, and therefore G is all fractional (g, f, n′, m)-critical deleted.

In all, this completes the proof of Theorem 9.

2.2 Proof of Theorem 7

In this part, we mainly present the detailed proof of Theorem 7.

Suppose G satisfies the hypothesis of Theorem 7, but is not an all fractional (g, f, n′, m)-critical deleted graph. Then via Theorem 10 and the fact that ∑xTdH(x) − eH(T, S) ≤ 2m for any HE(G) with m edges, there is a non-disjoint subset S, TV(G) satisfying

$a|S|−b|T|+∑x∈TdG−S(x)−an′−2m≤a(|S|−n′)+∑x∈T(dG−S(x)−dH(x)+eH(x,S)−b)≤g(S−U)+∑x∈T(dG−S(x)−dH(x)+eH(x,S)−f(x))≤−1.$(4)

S and T are chosen such that |T| is minimum. The subsets S and T ∖ {x} satisfy (4), when there is a xT which can satisfy dGS(x) ≥ g(x), This is conflicted with the selection rule of S and T. It infers that dGS(x) ≤ g(x) − 1 ≤ b − 1 for any xT.

Note that |T| ≠ ∅, otherwise we have ∑xTdH(x) − eH(T, S) = 0 and a(|S| − n′) < 0, contradicting (4). We write NT(x) = NG(x) ∩ T and NT[x] = (NG(x) ∩ T) ∪ {x} for any vertex xT.

Let

$d1=min{dG−S(x)|x∈T}$

and x1 be a vertex in T with dGS(x1) = d1. If TNT[x1] ≠ ∅, we further set

$d2=min{dG−S(x)|x∈T−NT[x1]},$

and x2 is a vertex in TNT[x1] with dGS(x2) = d2. Hence, we have d1d2b − 1 and |NG(x1) ∪ NG(x2)| ≤ |S| + d1 + d2.

The following discussion is divided into two cases.

• Case 1

T = NT[x1].

Since $\begin{array}{}\delta \left(G\right)\ge \frac{\left(a+b-1{\right)}^{2}+4b}{4a}\end{array}$ + m + n′, |T| ≤ d1 + 1 ≤ b and d1 + |S| ≥ dG(x1) ≥ δ(G), by (4) and the definition of d1, we have

$an′+2m−1≥a|S|−b|T|+∑x∈TdG−S(x)≥a|S|−b|T|+d1|T|≥a(δ(G)−d1)−(b−d1)|T|≥a(δ(G)−d1)−(b−d1)(d1+1)=d12−(a+b−1)d1+aδ(G)−b≥(d1−a+b−12)2+an′+2m≥an′+2m,$

• Case 2

TNT[x1] ≠ ∅.

Let p = |NT[x1]|, then we get d1p − 1. In view of TNT[x1] ≠ ∅, we obtain |T| ≥ p + 1. By virtue of n − |S| − |T| ≥ 0, d1d2 ≤ 0, bd2 ≥ 1, and (4), we yield

$(n−|S|−|T|)(b−d2)≥a|S|−b|T|+∑x∈TdG−S(x)−2m−an′+1≥a|S|−b|T|+d1p+d2(|T|−p)−2m−an′+1=a|S|+(d1−d2)p+(d2−b)|T|−2m−an′+1≥a|S|+(d1−d2)(d1+1)+(d2−b)|T|−2m−an′+1.$

It reveals

$(n−|S|)(b−d2)≥a|S|+(d1−d2)(d1+1)−2m−an′+1.$

By arranging the above inequality, we infer

$n(b−d2)−(a+b−d2)|S|+(d2−d1)(d1+1)+2m+an′−1≥0$(5)

Since vertices x1 and x2 are non-adjacent, using the neighborhood condition of the Theorem 7, we deduce

$bn+an′a+b≤|NG(x1)∪NG(x2)|≤|S|+d1+d2,$

which implies

$|S|≥bn+an′a+b−(d1+d2).$(6)

In terms of (5), (6), d1d2b − 1, and n$\begin{array}{}\frac{2\left(a+b\right)\left(a+b+m-1\right)}{a}\end{array}$ + n′, we get

$0≤n(b−d2)−(a+b−d2)|S|+(d2−d1)(d1+1)+2m+an′−1≤n(b−d2)−(a+b−d2)(bn+an′a+b−(d1+d2))+(d2−d1)(d1+1)+2m+an′−1=−and2a+b+(a+b−1)d2−d22+(a+b−1)d1−d12−(an′−d2an′a+b)+2m+an′−1≤−and1a+b+(a+b−1)d1−d12+(a+b−1)d1−d12+2m+d1an′a+b−1=−and1a+b+2(a+b)d1−2d12+2m+d1an′a+b−1≤2d1−2d12−1≤−1,$

Therefore, the desired result is proved.

2.3 Proof of Theorem 8

In the following part, we prove the second main result in this paper.

Assume that G satisfies the hypothesis of Theorem 8, but is not a fractional all (g, f, n′, m)-critical deleted graph. Then via Theorem 10 and the fact that ∑xTdH(x) − eH(T, S) ≤ 2m for any HE(G) with m edges, there is a non-disjoint subset S, TV(G) satisfying

$a|S|−b|T|+∑x∈TdG−S(x)−an′−2m≤a(|S|−n′)+∑x∈T(dG−S(x)−dH(x)+eH(x,S)−b)≤g(S−U)+∑x∈T(dG−S(x)−dH(x)+eH(x,S)−f(x))≤−1.$(7)

As depicted in the former subsection, S and T are selected such that |T| is the minimum. We get T ≠ ∅ and dGS(x) ≤ g(x) − 1 ≤ b − 1 for any xT.

The Claim below is firstly shown here.

Claim 1

G is assumed to be a graph of order n which satisfies the hypothesis of Theorem 8. After that, we obtain

$δ(G)≥(b−1)n+a+an′+2ma+b−1.$

Proof of Claim 1

Assume x to be a vertex of a graph G satisfying dG(x) = δ(G), and X = V(G) ∖ NG(x). Clearly, xNG(X), and hence NG(X) ≠ V(G). Combining this with the hypothesis of Theorem 8, we infer

$NG(X)≥(a+b−1)(n−1)an−a−an′−2m|X|.$

By virtue of |X| = nδ(G) and |NG(X)| ≤ n − 1, we deduce

$n−1≥|NG(X)|≥(a+b−1)(n−1)an−a−an′−2m|X|=(a+b−1)(n−1)an−a−an′−2m(n−δ(G)),$

which implies

$δ(G)≥(b−1)n+a+an′+2ma+b−1.$

Claim 1 is proved.

By setting d = min{dGS(x) : xT}, we have 0 ≤ db − 1. Select x1T with dGS(x1) = d. In light of δ(G) ≤ dG(x1) ≤ dGS(x1) + |S| = d + |S| and Claim 1, we yield

$|S|≥δ(G)−d≥(b−1)n+a+an′+2ma+b−1−d.$(8)

In what follows, we discuss three cases for the value of d.

• Case 1

2 ≤ db − 1.

By means of (8) and |S| + |T| ≤ n, we obtain

$a|S|+∑x∈TdG−S(x)−b|T|−an′−2m≥a|S|+d|T|−b|T|−an′−2m≥a|S|−(b−d)(n−|S|)−an′−2m=(a+b−d)|S|−(b−d)n−an′−2m≥(a+b−d)((b−1)n+a+an′+2ma+b−1−d)−(b−d)n−an′−2m.$

Let h(d) = (a + bd)( $\begin{array}{}\frac{\left(b-1\right)n+a+a{n}^{\prime }+2m}{a+b-1}\end{array}$d) − (bd)nan′ − 2m, then we have

$a|S|+∑x∈TdG−S(x)−b|T|−an′−2m≥h(d).$(9)

Using 2 ≤ db − 1 and n$\begin{array}{}\frac{\left(a+2b-3\right)\left(a+b-2\right)}{a}+\frac{a{n}^{\prime }}{a-1}\end{array}$ + m, we get

$h′(d)=−((b−1)n+a+an′+2ma+b−1−d)−(a+b−d)+n=2d+an−a−an′−2ma+b−1−a−b≥4+an−a−an′−2ma+b−1−a−b>0.$

Combining this with 2 ≤ db − 1, we infer

$h(d)≥f(2).$(10)

By virtue of (7), (9) and (10), we derive

$an′+2m−1≥h(d)≥h(2)=(a+b−2)((b−1)n+a+an′+2ma+b−1−2)−(b−2)n,$

which implies

$n≤(a+2b−3)(a+b−2)−1a+2ma+n′<(a+2b−3)(a+b−2)a+an′a−1+m,$

which conflicts with n$\begin{array}{}\frac{\left(a+2b-3\right)\left(a+b-2\right)}{a}+\frac{a{n}^{\prime }}{a-1}\end{array}$ + m.

• Case 2

d = 1.

In this case, we first prove the following two claims.

Claim 2

|T| ≤ $\begin{array}{}⌊\frac{\left(an-a-a{n}^{\prime }-2m\right)n}{\left(a+b-1\right)\left(n-1\right)}⌋\end{array}$.

Proof of Claim 2

Assume that |T| ≥ $\begin{array}{}⌊\frac{\left(an-a-a{n}^{\prime }-2m\right)n}{\left(a+b-1\right)\left(n-1\right)}⌋\end{array}$ + 1. Followed from dGS(x1) = d = 1, we have

$|T∖NG(x1)|≥|T|−1≥⌊(an−a−an′−2m)n(a+b−1)(n−1)⌋.$(11)

In view of (11) and the hypothesis of Theorem 8, we obtain

$NG(T∖NG(x1))=V(G).$(12)

On the other hand, it is clear that

$x1∉NG(T∖NG(x1)),$

which conflicts with (12).

Claim 3

|T| ≤ $\begin{array}{}\frac{an-a-a{n}^{\prime }-2m}{a+b-1}\end{array}$.

Proof of Claim 3

Assume that |T| > $\begin{array}{}\frac{an-a-a{n}^{\prime }-2m}{a+b-1}\end{array}$ . In terms of (8) and d = 1, we yield

$n≥|S|+|T|>(b−1)n+a+an′+2ma+b−1−1+an−a−an′−2ma+b−1=n−1,$

which means

$|S|+|T|=n.$(13)

According to (7), (13) and Claim 2, we get

$an′+2m−1≥a|S|+∑x∈TdG−S(x)−b|T|≥a|S|+|T|−b|T|=a(n−|T|)−(b−1)|T|=an−(a+b−1)|T|≥an−(a+b−1)(an−a−an′−2m)n(a+b−1)(n−1)=an−(an−a−an′−2m)nn−1≥an′+2m,$

a contradiction. Hence the Claim 3 is hold.

Now, we set t′ = |{x : xT, dGS(x) = 1}|. It is obvious that t′ ≥ 1 and |T| ≥ t′. In light of (8), d = 1 and Claim 3, we obtain

$a|S|+∑x∈TdG−S(x)−b|T|−an′−2m≥a|S|+2|T|−t′−b|T|−an′−2m=a|S|−(b−2)|T|−t′−an′−2m≥a((b−1)n+a+an′+2ma+b−1−1)−an′−2m−(b−2)an−a−an′−2ma+b−1−t′=an−a−an′−2ma+b−1−t′≥|T|−t′≥0,$

which conflicts with (7).

• Case 3

d = 0.

Let r = |{x : xT, dGS(x) = 0}| and Y = V(G) ∖ S. Obviously, we have r ≥ 1 and NG(Y) ≠ V(G). Combining these with the hypothesis of Theorem 8, we derive

$n−r≥|NG(Y)|≥(a+b−1)(n−1)an−a−an′−2m|Y|=(a+b−1)(n−1)an−a−an′−2m(n−|S|),$

which implies,

$|S|≥n−(n−r)(an−a−an′−2m)(a+b−1)(n−1).$(14)

In view of n$\begin{array}{}\frac{\left(a+2b-3\right)\left(a+b-2\right)}{a}+\frac{a{n}^{\prime }}{a-1}\end{array}$ + m, we ensure that

$an−a−an′−2mn−1>1.$(15)

By means of (7), (14), (15) and |S| + |T| ≤ n, we infer

$an′+2m−1≥a|S|+∑x∈TdG−S(x)−b|T|≥a|S|−(b−1)|T|−r≥a|S|−(b−1)(n−|S|)−r=(a+b−1)|S|−(b−1)n−r≥(a+b−1)(n−(n−r)(an−a−an′−2m)(a+b−1)(n−1))−(b−1)n−r=an−(n−r)(an−a−an′−2m)n−1−r≥an−(n−1)(an−a−an′−2m)n−1−1=an−(an−a−an′−2m)−1=an′+2m+a−1>an′+2m,$

In conclusion, we complete the proof of Theorem 8.

3 Sharpness

The most likely in the sense that we can’t replace $\begin{array}{}\frac{bn+a{n}^{\prime }}{a+b}\end{array}$ by $\begin{array}{}\frac{bn+a{n}^{\prime }}{a+b}\end{array}$ − 1 is the lower bound on the condition |NG(x) ∪ NG(y)| ≥ $\begin{array}{}\frac{bn+a{n}^{\prime }}{a+b}\end{array}$ in Theorem 7. We present this by constructing a graph G = (bt + n′)K1 ∨ (at + 1)K1, where a and b are two nonnegative integers with 2 ≤ ab and t is a large sufficiently positive integer. Obviously,

$|V(G)|=n=(a+b)t+n′+1$

and

$bn+an′a+b>|NG(x)∪NG(y)|=bt+n′>bn+an′a+b−1$

for any non-adjacent vertices x, yV((at + 1)K1). Set S = V((bt + n′)K1), T = V((at + 1)K1). Let US with |U| = n′. Let H be a subgraph of T with m edges. Obviously, |S| = bt + n′, |T| = at + 1, ∑xTdGS(x) = 0 and ∑xTdH(x) − eH(S, T) = 0. Furthermore, let g(x) = a for any xS, and f(x) = b for any xT. Hence, we obtain

$g(S−U)+∑x∈T(dG−S(x)−dH(x)+eH(x,S)−f(x))=g(S−U)−f(T)=abt−b(at+1)=−b<0.$

By Theorem 10, G is not all fractional (g, f, n′, m)-critical deleted.

The bound of the hypothesis in Theorem 8 is the best possible in some sense, i.e., it can’t be replaced by NG(X) = V(G) or |NG(X)| ≥ $\begin{array}{}\frac{\left(a+b-1\right)\left(n-1\right)}{an-a-a{n}^{\prime }-2m}\end{array}$ |X| for any XV(G) (i.e., the restriction condition on |X| is necessary). We explain this by constructing a graph G as follows. Let V(G) = ST with ST = ∅, |S| = (b − 1)t + n′ and |T| = at + 1, and T = {x1, x2, ⋯, x2t′}, where n′ ≥ 0, ba ≥ 2 are integers, a and t are odd integers. Obviously, 2t′ = at + 1. For yS, we write NG(y) = V(G) ∖ {y}. For xT, we write NG(x) = S ∪ {x′}, where {x, x′} = {x2i−1, x2i} for some 1 ≤ it′. It is obvious that n = (b − 1)t + n′ + at + 1. Next, we show that NG(X) = V(G) or |NG(X)| ≥ $\begin{array}{}\frac{\left(a+b-1\right)\left(n-1\right)}{an-a-a{n}^{\prime }-2m}\end{array}$ |X| holds for any XV(G). Apparently, NG(X) = V(G) if |XS| ≥ 2, or |XS| = 1 and |XT| ≥ 1 for any XV(G). If |X| = 1 and XS, then |NG(X)| = |V(G)| − 1 = n − 1 > $\begin{array}{}\frac{\left(a+b-1\right)\left(n-1\right)}{at\left(a+b-1\right)}=\frac{\left(a+b-1\right)\left(n-1\right)}{an-a-a{n}^{\prime }-2m}=\frac{\left(a+b-1\right)\left(n-1\right)}{an-a-a{n}^{\prime }-2m}\end{array}$ |X|. Thus, we may suppose XT. Note that |NG(X)| = |S| + |X| = (b − 1)t + n′ + |X|. Thus, |NG(X)| ≥ $\begin{array}{}\frac{\left(a+b-1\right)\left(n-1\right)}{an-a-a{n}^{\prime }-2m}\end{array}$ |X| holds if and only if (b − 1)t + n′ + |X| ≥ $\begin{array}{}\frac{\left(a+b-1\right)\left(n-1\right)}{b\left(n-1\right)-b{n}^{\prime }-2m}\end{array}$ |X|, which is equivalent to |X| ≤ at. Clearly, |NG(X)| ≥ $\begin{array}{}\frac{\left(a+b-1\right)\left(n-1\right)}{b\left(n-1\right)-b{n}^{\prime }-2m}\end{array}$ |X| holds for any XT with XT. If X = T, then NG(X) = V(G). As a result, NG(X) = V(G) or |NG(X)| ≥ $\begin{array}{}\frac{\left(a+b-1\right)\left(n-1\right)}{an-a-a{n}^{\prime }-2m}\end{array}$ |X| holds for any XV(G). Then, we show that G is not all fractional (g, f, n′, m)-critical deleted. For above S and T, it is obvious that |S| > n′ and dGS(x) = 1 for each xT. Let g(x) = a for any xS, and f(x) = b for any xT. Let H be a subgraph of T with m edges. Therefore, in terms of ∑xTdGS(x) − ∑xTdH(x) + eH(S, T) ≥ 0 holds for any H, we obtain

$g(S−U)+∑x∈T(dG−S(x)−dH(x)+eH(x,S)−f(x))=a(|S|−n′)+(∑x∈TdG−S(x)−∑x∈TdH(x)+eH(S,T))−b|T|≥a(b−1)t−b(at+1)=−at−bt<0.$

It follows from Theorem 10 that G is not all fractional (g, f, n′, m)-critical deleted.

4 Conclusions

In recent years, the problem of fractional factor in graphs has raised much attention in the field of graph theory and computer networks. In this paper, we consider the theoretical problems in data transmission networks when some sites and channels are not available in the certain time. The relationship between neighborhood conditions and a graph to be all fractional (g, f, n’, m)- critical deleted is discussed. Two sufficient neighborhood conditions are obtained, and the sharpness of conditions is presented. The theoretical conclusions we yield in this paper have potential applications in network design and information transmission.

Acknowledgement

We thank the reviewers for their insightful comments in the improvement of the paper. The work has been partially supported by Postdoctoral Research Grant of China (2017M621690), National Science Foundation of China (11401519), postdoctoral research grant in Jiangsu province (1701128B).

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Accepted: 2018-05-15

Published Online: 2018-08-20

Conflict of InterestConflict of Interests: The authors hereby declare that there is no conflict of interests regarding the publication of this paper.

Citation Information: Open Physics, Volume 16, Issue 1, Pages 544–553, ISSN (Online) 2391-5471,

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