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BINDING NUMBERS AND FRACTIONAL (g, f, n)-CRITICAL GRAPHS
ZHOU, SIZHONG,SUN, ZHIREN The Korean Society for Computational and Applied M 2016 Journal of applied mathematics & informatics Vol.34 No.5
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.
Binding numbers and fractional $(g,f,n)$-critical graphs
Sizhong Zhou,Zhiren Sun 한국전산응용수학회 2016 Journal of applied mathematics & informatics Vol.34 No.5
Let $G$ be a graph, and let $g,f$ be two nonnegative integer-valued functions defined on $V(G)$ with $g(x)\leq f(x)$ for each $x\in V(G)$. 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.
REMARKS ON NEIGHBORHOODS OF INDEPENDENT SETS AND (a, b, k)-CRITICAL GRAPHS
Zhou, Sizhong,Sun, Zhiren,Xu, Lan The Korean Society for Computational and Applied M 2013 Journal of applied mathematics & informatics Vol.31 No.5
Let $a$ and $b$ be two even integers with $2{\leq}a<b$, and let k be a nonnegative integer. Let G be a graph of order $n$ with $n{\geq}\frac{(a+b-1)(a+b-2)+bk-2}{b}$. A graph G is called an ($a,b,k$)-critical graph if after deleting any $k$ vertices of G the remaining graph of G has an [$a,b$]-factor. In this paper, it is proved that G is an ($a,b,k$)-critical graph if $${\mid}N_G(X){\mid}>\frac{(a-1)n+{\mid}X{\mid}+bk-2}{a+b-1}$$ for every non-empty independent subset X of V (G), and $${\delta}(G)>\frac{(a-1)n+a+b+bk-3}{a+b-1}$$. Furthermore, it is shown that the result in this paper is best possible in some sense.
REMARKS ON NEIGHBORHOODS OF INDEPENDENT SETS AND (a, b, k)-CRITICAL GRAPHS
Sizhong Zhou,Zhiren Sun,Lan Xu 한국전산응용수학회 2013 Journal of applied mathematics & informatics Vol.31 No.5
Let a and b be two even integers with 2 ≤ a < b, and let k be a nonnegative integer. Let G be a graph of order n with n ≥(a+b-1)(a+b-2)+bk-2 b . A graph G is called an (a, b, k)-critical graph if afterdeleting any k vertices of G the remaining graph of G has an [a, b]-factor. In this paper, it is proved that G is an (a, b, k)-critical graph if |NG(X)| >(a - 1)n + |X| + bk - 2a + b - 1 for every non-empty independent subset X of V (G), and δ(G) >(a -1)n + a + b + bk - 3a + b - 1. Furthermore, it is shown that the result in this paper is best possible insome sense.