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NOTE ON CONNECTED (g, f)-FACTORS OF GRAPHS
Sizhong Zhou,Jiancheng Wu,Quanru Pan 한국전산응용수학회 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 intergers with 1≤ a < b and b ≥ 3, and let g and f be two integer-valued functions defined on V (G) such that a ≤ g(x) < f(x) ≤ b for each x ∈ V(G)and f(V (G))−V (G) even. If [수식], then G has a connected (g,f)-factor.
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.
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.
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
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].
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).
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 $\delta(G)$ > $\frac{(a-2)n+2{\alpha}(G)-1)}{a+b-4}$.
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.
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.