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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.
MINIMUM DEGREE AND INDEPENDENCE NUMBER FOR THE EXISTENCE OF HAMILTONIAN [a, b]-FACTORS
Zhou, Sizhong,Pu, Bingyuan The Korean Society for Computational and Applied M 2010 Journal of applied mathematics & informatics Vol.28 No.1
Let a and b be nonnegative integers with 2 $\leq$ a < b, and let G be a Hamiltonian graph of order n with n > $\frac{(a+b-5)(a+b-3)}{b-2}$. An [a, b]-factor F of G is called a Hamiltonian [a, b]-factor if F contains a Hamiltonian cycle. In this paper, it is proved that G has a Hamiltonian [a, b]-factor if $\delta(G)\;\geq\;\frac{(a-1)n+a+b-3)}{a+b-3}$ and $\delta(G)$ > $\frac{(a-2)n+2{\alpha}(G)-1)}{a+b-4}$.
RANDOMLY ORTHOGONAL FACTORIZATIONS OF (0,mf − (m− 1)r)-GRAPHS
Sizhong Zhou,Minggang Zong 대한수학회 2008 대한수학회지 Vol.45 No.6
Let G be a graph with vertex set V (G) and edge set E(G), and let g, f be two nonnegative integer-valued functions defined on V (G) such that g(x) ≤ f(x) for every vertex x of V (G). We use dG(x) to denote the degree of a vertex x of G. A (g, f)-factor of G is a spanning subgraph F of G such that g(x) ≤ dF (x) ≤ f(x) for every vertex x of V (F). In particular, G is called a (g, f)-graph if G itself is a (g, f)-factor. A (g, f)- factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let F = {F₁, F₂, . . . , Fm} be a factorization of G and H be a subgraph of G with mr edges. If Fi, 1 ≤ i ≤ m, has exactly r edges in common with H, we say that F is r-orthogonal to H. If for any partition {A₁,A₂, . . . ,Am} of E(H) with |Ai| = r there is a (g, f)-factorization F = {F₁, F₂, . . . , Fm} of G such that Ai ⊆ E(Fi), 1 ≤ i ≤ m, then we say that G has (g, f)- factorizations randomly r-orthogonal to H. In this paper it is proved that every (0,mf − (m − 1)r)-graph has (0, f)-factorizations randomly r-orthogonal to any given subgraph with mr edges if f(x) ≥ 3r − 1 for any x ∈ V (G). Let G be a graph with vertex set V (G) and edge set E(G), and let g, f be two nonnegative integer-valued functions defined on V (G) such that g(x) ≤ f(x) for every vertex x of V (G). We use dG(x) to denote the degree of a vertex x of G. A (g, f)-factor of G is a spanning subgraph F of G such that g(x) ≤ dF (x) ≤ f(x) for every vertex x of V (F). In particular, G is called a (g, f)-graph if G itself is a (g, f)-factor. A (g, f)- factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let F = {F₁, F₂, . . . , Fm} be a factorization of G and H be a subgraph of G with mr edges. If Fi, 1 ≤ i ≤ m, has exactly r edges in common with H, we say that F is r-orthogonal to H. If for any partition {A₁,A₂, . . . ,Am} of E(H) with |Ai| = r there is a (g, f)-factorization F = {F₁, F₂, . . . , Fm} of G such that Ai ⊆ E(Fi), 1 ≤ i ≤ m, then we say that G has (g, f)- factorizations randomly r-orthogonal to H. In this paper it is proved that every (0,mf − (m − 1)r)-graph has (0, f)-factorizations randomly r-orthogonal to any given subgraph with mr edges if f(x) ≥ 3r − 1 for any x ∈ V (G).
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.
BINDING NUMBER CONDITIONS FOR (a, b, k)-CRITICAL GRAPHS
Sizhong Zhou 대한수학회 2008 대한수학회보 Vol.45 No.1
Let G be a graph, and let a, b, k be integers with 0 ≤ a ≤ b, k ≥ 0. Then 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, the relationship between binding number bind(G) and (a, b, k)-critical graph is discussed, and a binding number condition for a graph to be (a, b, k)-critical is given.
Degree conditions and fractional k-factors of graphs
Sizhong Zhou 대한수학회 2011 대한수학회보 Vol.48 No.2
Let k≥1 be an integer, and let G be a 2-connected graph of order n≥,max{7, 4k+1}, and the minimum degree δ(G)≥ k+1. In this paper, it is proved that G has a fractional k-factor excluding any given edge if G satisfies max{d_G(x),d_G(y)}≥n/2 for each pair of nonadjacent vertices x,y of G. Furthermore, it is showed that the result in this paper is best possible in some sense.
NOTE ON CONNECTED (g, f)-FACTORS OF GRAPHS
Zhou, Sizhong,Wu, Jiancheng,Pan, Quanru The Korean Society for Computational and Applied M 2010 Journal of applied mathematics & informatics Vol.28 No.3
In this note we present a short proof of the following result by Zhou, Liu and Xu. Let G be a graph of order n, and let a and b be two integers with 1 $\leq$ a < b and b $\geq$ 3, and let g and f be two integer-valued functions defined on V(G) such that a $\leq$ g(x) < f(x) $\leq$ b for each $x\;{\in}\;V(G)$ and f(V(G)) - V(G) even. If $n\;{\geq}\;\frac{(a+b-1)^2+1}{a}$ and $\delta(G)\;{\geq}\;\frac{(b-1)n}{a+b-1}$,then G has a connected (g, f)-factor.
MINIMUM DEGREE AND INDEPENDENCE NUMBER FOR THE EXISTENCE OF HAMILTONIAN [a, b]-FACTORS
Sizhong Zhou,Bingyuan Pu 한국전산응용수학회 2010 Journal of applied mathematics & informatics Vol.28 No.1
Let a and b be nonnegative integers with 2 ≤ a < b, and let, G be a Hamiltonian graph of order n with n >[수식]. An [a, b]-factor F of G is called a Hamiltonian [a, b]-factor if F contains a Hamiltonian cycle. In this paper, it is proved that G has a Hamiltonian [a, b]-factor if δ(G) ≥[수식] and δ(G) >[수식].
BINDING NUMBER CONDITIONS FOR (a, b, k)-CRITICAL GRAPHS
Zhou, Sizhong Korean Mathematical Society 2008 대한수학회보 Vol.45 No.1
Let G be a graph, and let a, b, k be integers with $0{\leq}a{\leq}b,k\geq0$. Then 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, the relationship between binding number bind(G) and (a, b, k)-critical graph is discussed, and a binding number condition for a graph to be (a, b, k)-critical is given.