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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)
  • Received : 2015.11.28
  • Accepted : 2016.03.24
  • Published : 2016.09.30

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.

Keywords

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.

References

  1. J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, GTM-244, Berlin: Springer, 2008.
  2. O. Fourtounelli and P. Katerinis, The existence of k-factors in squares of graphs, Discrete Math. 310 (2010), 3351-3358. https://doi.org/10.1016/j.disc.2010.07.024
  3. P. Katerinis and D.R. Woodall, Binding numbers of graphs and the existence of k-factors, Quart. J. Math. Oxford 38 (1987), 221-228. https://doi.org/10.1093/qmath/38.2.221
  4. K. Kimura, f-factors, complete-factors, and component-deleted subgraphs, Discrete Math. 313 (2013), 1452-1463. https://doi.org/10.1016/j.disc.2013.03.009
  5. M. Kouider and S. Ouatiki, Sufficient condition for the existence of an even [a, b]-factor in graph, Graphs Combin. 29 (2013), 1051-1057.
  6. R. Kužel, K. Ozeki and K. Yoshimoto, 2-factors and independent sets on claw-free graphs, Discrete Math. 312 (2012), 202-206. https://doi.org/10.1016/j.disc.2011.08.020
  7. G. Liu and L. Zhang, Characterizations of maximum fractional (g, f)-factors of graphs, Discrete Appl. Math. 156 (2008), 2293-2299. https://doi.org/10.1016/j.dam.2007.10.016
  8. G. Liu and L. Zhang, Toughness and the existence of fractional k-factors of graphs, Discrete Math. 308 (2008), 1741-1748. https://doi.org/10.1016/j.disc.2006.09.048
  9. S. Liu, On toughness and fractional (g, f, n)-critical graphs, Inform. Process. Lett. 110 (2010), 378-382. https://doi.org/10.1016/j.ipl.2010.03.005
  10. H. Lu, Regular graphs, eigenvalues and regular factors, J. Graph Theory 69 (2012), 349-355. https://doi.org/10.1002/jgt.20581
  11. H. Lu, Simplified existence theorems on all fractional [a, b]-factors, Discrete Appl. Math. 161 (2013), 2075-2078. https://doi.org/10.1016/j.dam.2013.02.006
  12. D.R. Woodall, The binding number of a graph and its Anderson number, J. Combin. Theory ser. B 15 (1973), 225-255. https://doi.org/10.1016/0095-8956(73)90038-5
  13. S. Zhou, A new neighborhood condition for graphs to be fractional (k,m)-deleted graphs, Appl. Math. Lett. 25 (2012), 509-513. https://doi.org/10.1016/j.aml.2011.09.048
  14. S. Zhou, Independence number, connectivity and (a, b, k)-critical graphs, Discrete Math. 309 (2009), 4144-4148. https://doi.org/10.1016/j.disc.2008.12.013
  15. S. Zhou and Q. Shen, On fractional (f, n)-critical graphs, Inform. Process. Lett. 109 (2009), 811-815. https://doi.org/10.1016/j.ipl.2009.03.026
  16. S. Zhou, Z. Sun and H. Ye, A toughness condition for fractional (k,m)-deleted graphs, Inform. Process. Lett. 113 (2013), 255-259. https://doi.org/10.1016/j.ipl.2013.01.021