BINDING NUMBERS AND FRACTIONAL (g, f, n)-CRITICAL GRAPHS

• ZHOU, SIZHONG (School of Mathematics and Physics, Jiangsu University of Science and Technology) ;
• SUN, ZHIREN (School of Mathematical Sciences, Nanjing Normal University)
• 투고 : 2015.11.28
• 심사 : 2016.03.24
• 발행 : 2016.09.30

초록

Let G be a graph, and let g, f be two nonnegative integer-valued functions defined on V (G) with g(x) ≤ f(x) for each x ∈ V (G). A graph G is called a fractional (g, f, n)-critical graph if after deleting any n vertices of G the remaining graph of G admits a fractional (g, f)-factor. In this paper, we obtain a binding number condition for a graph to be a fractional (g, f, n)-critical graph, which is an extension of Zhou and Shen's previous result (S. Zhou, Q. Shen, On fractional (f, n)-critical graphs, Inform. Process. Lett. 109(2009)811-815). Furthermore, it is shown that the lower bound on the binding number condition is sharp.

1. Introduction

The graphs considered in this paper are finite, undirected and simple, and see  for all notation and terminology not explained here.

Let G be a graph. We denote its vertex set and edge set by V (G) and E(G), respectively. The degree dG(v) of a vertex v ∈ V (G) is the number of edges of G incident with v. Set δ(G) = min{dG(v) : v ∈ V (G)}. The neighborhood of a vertex v in G is the set NG(v) = {u ∈ V (G) : vu ∈ E(G)}. For X ⊆ V (G), we write NG(X) for the union of NG(v) for each v ∈ X and denote by G[X] the subgraph of G induced by X. Set G − X = G[V (G) \ X]. The binding number of a graph G is denoted by bind(G) and it is defined as

Let g and f be two integer-valued functions defined on V (G) with 0 ≤ g(x) ≤ f(x) for each x ∈ V (G). A (g, f)-factor of a graph G is a spanning subgraph F of G satisfying g(x) ≤ dF (x) ≤ f(x) for each x ∈ V (G). A fractional (g, f)-factor of a graph G is a function h that assigns to each edge of G a number in [0, 1], so that for any x ∈ V (G) we have where (the sum is taken over all edges incident to x) is a fractional degree of x in G. A fractional (f, f)-factor is abbreviated to a fractional f-factor. A fractional (g, f)-factor is a fractional [a, b]-factor if g(x) = a and f(x) = b for each x ∈ V (G). If a = b = k, then a fractional [k, k]-factor is said to be a fractional k-factor. A graph G is called a fractional (g, f, n)-critical graph if after deleting any n vertices of G the remaining graph of G admits a fractional (g, f)-factor. A fractional (f, f, n)-critical graph is abbreviated to a fractional (f, n)-critical graph. If g(x) = a and f(x) = b for each x ∈ V (G), then a fractional (g, f, n)-critical graph is said to be a fractional (a, b, n)-critical graph. A fractional (f, n)-critical graph is a fractional (k, n)-critical graph if f(x) = k for each x ∈ V (G).

Many results on factors [2-6,10,14] and fractional factors [7,8,11,13,16] of graphs are known.

Zhou and Shen  proved the following theorem, which shows the the relationship between binding number and fractional (f, n)-critical graphs.

Theorem 1 (). Let G be a graph of order p, and let a, b and n be nonnegative integers such that 2 ≤ a ≤ b, and let f be an integer-valued function defined on V (G) such that a ≤ f(x) ≤ b for each x ∈ V (G). If then G is fractional (f, n)-critical.

Liu extended a fractional (f, n)-critical graph to a fractional (g, f, n)-critical graph and obtained a toughness condition for the existence of fractional (g, f, n)-critical graphs in .

Theorem 2 (). Let G be a graph and let g, f be two nonnegative integer-valued functions defined on V (G) satisfying a ≤ g(x) ≤ f(x) ≤ b with 1 ≤ a ≤ b and b ≥ 2 for all x ∈ V (G), where a, b are positive integers. If then G is a fractional (g, f, n)-critical graph, where n is a positive integer with |V (G)| ≥ n + 1.

In this paper, we proceed to investigate the fractional (g, f, n)-critical graphs and obtain a binding number condition for the existence of fractional (g, f, n)-critical graphs, which is an extension of Theorem 1. Our main result is the following theorem.

Theorem 3. Let a, b, r and n be four nonnegative integers with 2 ≤ a ≤ b − r, and let G be a graph of order p with and let g, f be two integer-valued functions defined on V (G) with a ≤ g(x) ≤ f(x) − r ≤ b − r for each x ∈ V (G). If then G is fractional (g, f, n)-critical.

If n = 0 in Theorem 3, we obtain the following corollary.

Corollary 1. Let a, b and r be three nonnegative integers with 2 ≤ a ≤ b−r, and let G be a graph of order p with and let g, f be two integer-valued functions defined on V (G) with a ≤ g(x) ≤ f(x) − r ≤ b − r for each x ∈ V (G). If then G has a fractional (g, f)-factor.

If r = 0 in Theorem 3, then we have the following corollary.

Corollary 2. Let a, b and n be three nonnegative integers with 2 ≤ a ≤ b, and let G be a graph of order p with and let g, f be two integer-valued functions defined on V (G) with a ≤ g(x) ≤ f(x) ≤ b for each x ∈ V (G). If then G is fractional (g, f, n)-critical.

2. The Proof of Theorem 3

The purpose of this section is to prove Theorem 3. For the proof of Theorem 3, we need the following lemmas.

Lemma 2.1 (). Let G be a graph, and let n be a nonnegative integer, and let g, f be two integer-valued functions defined on V (G) with 0 ≤ g(x) ≤ f(x) for each x ∈ V (G). Then G is fractional (g, f, n)-critical if and only if for any subset S of V (G) with |S| ≥ n

where T = {x : x ∈ V (G) \ S, dG−S(x) ≤ g(x)}, dG−S(T) = Σx∈T dG−S(x) and fn(S) = max{f(U) : U ⊆ S, |U| = n}.

Lemma 2.2. Let G be a graph of order p, and let a, b, r and n are four nonnegative integers with 1 ≤ a ≤ b−r, and let g, f be two integer-valued functions defined on V (G) satisfying a ≤ g(x) ≤ f(x) − r ≤ b − r for each x ∈ V (G). If then G is fractional (g, f, n)-critical.

Proof. Suppose that G satisfies the hypothesis of Lemma 2.2, but it is not fractional (g, f, n)-critical. Then according to Lemma 2.1, there exists some subset S of V (G) with |S| ≥ n satisfying

where T = {x : x ∈ V (G) \ S, dG−S(x) ≤ g(x)}, dG−S(T) = Σx∈T dG−S(x) and fn(S) = max{f(U) : U ⊆ S, |U| = n}.

Note that f(S) ≥ fn(S). If then by (1) we have fn(S) − 1 ≥ f(S) ≥ fn(S), a contradiction. Therefore, In the following, we define h = min{dG−S(x) : x ∈ T}. According to the definition of T, we have 0 ≤ h ≤ b−r.

We choose x1 ∈ T with dG−S(x1) = h. Thus, we obtain

As a consequence,

Note that fn(S) = max{f(U) : U ⊆ S, |U| = n} ≤ bn. And then using (1), (2) and |S| + |T| ≤ p, we obtain

Solving for δ(G), we obtain the following

Let Taking the derivative of F(h) with respect to h yields

For which implies that F(h) attains its maximum value at h = 0. Hence,

which contradicts The proof of Lemma 2.2 is complete. □

Lemma 2.3 (). Let c be a positive real, and let G be a graph of order p with bind(G) := β > c. Then

Proof of Theorem 3. Suppose that G satisfies the hypothesis of Theorem 3, but it is not fractional (g, f, n)-critical. Again, we apply Lemma 2.1, with the same notations and sets as defined in the proof of Lemma 2.2. In addition, we use β := bind(G) to simplify the notation below.

In the following, we need only to consider h = 0; for h ≥ 1, apply the same argument as in Lemma 2.2. Let Y = {x : x ∈ T, dG−S(x) = 0}. Obviously, Note that |NG(V (G) \ S)| ≤ p − |Y|. According to the definition of bind(G), we have

that is,

It follows from (1), (3), fn(S) ≤ bn and |S| + |T| ≤ p that

that is,

We may assume that β ≤ a + b − 1. Otherwise, by Lemma 2.3 and we have and Lemma 2.2 can be applied. Furthermore, we obtain by (4)

which implies

which contradicts This completes the proof of Theorem 3. □

3. Remark

In this section, we show that the condition in Theorem 3 is best possible.

Let a, b, r and n be four nonnegative integers such that 2 ≤ a = b − r, a + b + 2 + n is even and is an integer. We write 2l = a + b + 2 + n and Set Let g(x) and f(x) be two integer-valued functions defined on V (G) with g(x) ≡ a and f(x) ≡ b = a + r. We choose X = V (lK2). Then |NG(X \ x)| = p − 1 for each x ∈ X. Obviously, For S = V (Km) and T = V(lK2), we obtain

So by Lemma 2.1, G is not fractional (g, f, n)-critical.

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