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

Abstract

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 [1] 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 [15] proved the following theorem, which shows the the relationship between binding number and fractional (f, n)-critical graphs.

Theorem 1 ([15]). 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 [9].

Theorem 2 ([9]). 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 ([9]). 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 ([12]). 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.