On Vertex-Disjoint Complete Bipartite Subgraphs in a Bipartite Graph

 We show that, if G=(X,Y;E) is a bipartite graph with |X|=|Y|=4s and δ(G)≥4s−3 for any integer s≥2, then G contains four vertex-disjoint copies of K _s,s. This constitutes a partial answer to a conjecture in [4]. Received: March 27, 1996 Revised: March 7, 1997     

On Independent Cycles in a Bipartite Graph

 Let G=(V ₁,V ₂;E) be a bipartite graph with 2k≤m=|V ₁|≤|V ₂|=n, where k is a positive integer. We show that if the number of edges of G is at least (2k−1)(n−1)+m, then G contains k vertex-disjoint cycles, unless e(G)=(2k−1)(n−1)+m and G belongs to a known class of graphs. Received: December 9, 1998 Final version received: June 2, 1999     

On a Conjecture on Directed Cycles in a Directed Bipartite Graph

Let D = (V ₁, V ₂; A) be a directed bipartite graph with |V ₁| = |V ₂| = n ≥ 2. Suppose that d _D (x) + d _D (y) ≥ 3n for all x ∈ V ₁ and y ∈ V ₂. Then, with one exception, D contains two vertex-disjoint directed cycles of lengths 2n ₁ and 2n ₂, respectively, for any positive integer partition n = n ₁ + n ₂. This proves a conjecture proposed in [9].     

均衡二部图中含2k条指定边的k个独立圈及2-因子

得到了对于二部图G=(V_1,V_2;E),当|V_1|=|V_2|=n≥2k+1时的结果:对G中任意2k条独立边e_1,e_1~*,…,e_k,e_k~*,G中一定存在k个独立的4-圈C_1,C_2,…,C_k,使得对任意i∈{1,2,…,k}有{e_i,e_i~*}E(C_i).并在此基础上进一步证明了当|V_1|=|V_2|=n≥3k时若对任意两顶点x∈V_1,y∈V_2,都有d(x)+d(y)≥2n-k+1成立,则G有一个2-因子含有k+1个独立圈C_1,C_2,…,C_(k+1)使得对任意i∈{1,2,…,k}有{e_i,e_i~*}E(C_i)且|C_i|=4.     

Multicolored Bipartite Ramsey Numbers of Large Cycles

For an integer r≥ 2 and bipartite graphs H_i,where 1 ≤i≤r,the bipartite Ramsey number br(H₁,H₂,…,H_r) is the minimum integer N such that any r-edge coloring of the complete bipartite graph K_N,N contains a monochromatic subgraph isomorphic to H_i in color i for some 1≤i≤r.We show that if ■     

Vertex-Disjoint Cycles Containing Specified Edges

Dirac and Ore-type degree conditions are given for a graph to contain vertex disjoint cycles each of which contains a previously specified edge. One set of conditions is given that imply vertex disjoint cycles of length at most 4, and another set of conditions are given that imply the existence of cycles that span all of the vertices of the graph (i.e. a 2-factor). The conditions are shown to be sharp and give positive answers to conjectures of Enomoto in [3] and Wang in [5]. Revised: July 28, 1999     

Spanning Cyclic Subdivisions of Vertex-Disjoint Cycles and Chorded Cycles in Graphs

Let G be a graph on n ≥ 3 vertices and H be a subgraph of G such that each component of H is a cycle with at most one chord. In this paper we prove that if the minimum degree of G is at least n/2, then G contains a spanning subdivision of H such that only non-chord edges of H are subdivided. This gives a new generalization of the classical result of Dirac on the existence of Hamilton cycles in graphs.     

The Local Structure of a Bipartite Distance-regular Graph

In this paper, we consider a bipartite distance-regular graph Γ = (X, E) with diameter d ≥ 3. We investigate the local structure ofΓ , focusing on those vertices with distance at most 2 from a given vertex x. To do this, we consider a subalgebra R = R(x) ofMat_{0307a0x.gif X}(C), where 0307a1x.gifX denotes the set of vertices in X at distance 2 from x. R is generated by matrices Ã, 0307a2x.gif J, and 0307a3x.gif D defined as follows. For all y, z 0307a4x.gif X, the (y,z )-entry of à is 1 if y, z are at distance 2, and 0 otherwise. The (y, z)-entry of 0307a5x.gif J equals 1, and the (y,z )-entry of 0307a6x.gif D equals the number of vertices of X adjacent to each ofx , y, and z. We show that R is commutative and semisimple, with dimension at least 2. We assume thatR is one of 2, 3, or 4, and explore the combinatorial implications of this. We are motivated by the fact that if Γ has a Q-polynomial structure, thenR ≤ 4.     

Cycles in 2-Factors of Balanced Bipartite Graphs

In the study of hamiltonian graphs, many well known results use degree conditions to ensure sufficient edge density for the existence of a hamiltonian cycle. Recently it was shown that the classic degree conditions of Dirac and Ore actually imply far more than the existence of a hamiltonian cycle in a graph G, but also the existence of a 2-factor with exactly k cycles, where . In this paper we continue to study the number of cycles in 2-factors. Here we consider the well-known result of Moon and Moser which implies the existence of a hamiltonian cycle in a balanced bipartite graph of order 2n. We show that a related degree condition also implies the existence of a 2-factor with exactly k cycles in a balanced bipartite graph of order 2n with . Revised: May 7, 1999     

赋模糊权二部图的最佳匹配

本文建立了赋模糊数为边权的二部图中，据模糊决策来求解最佳匹配的网络模型，并给出了这一模型的有效算法     

Neighborhood Unions and Hamilton Cycles in Bipartite Graphs

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Forbidden Triples Containing a Complete Graph and a Complete Bipartite Graph of Small Order

For a graph G and a set \({\mathcal{F}}\) of connected graphs, G is said be \({\mathcal{F}}\) -free if G does not contain any member of \({\mathcal{F}}\) as an induced subgraph. We let \({\mathcal{G} _{3}(\mathcal{F})}\) denote the set of all 3-connected \({\mathcal{F}}\) -free graphs. This paper is concerned with sets \({\mathcal{F}}\) of connected graphs such that \({\mathcal{F}}\) contains no star, \({|\mathcal{F}|=3}\) and \({\mathcal{G}_{3}(\mathcal{F})}\) is finite. Among other results, we show that for a connected graph T( ≠ K ₁) which is not a star, \({\mathcal{G}_{3}(\{K_{4},K_{2,2},T\})}\) is finite if and only if T is a path of order at most 6.     

关于k-消去二分图的一些结果

图 G的一个 k-正则支撑子图称为 G的 k-因子 ,若对 G的任一边 e,图 G- e总存在一个 k-因子 ,则称 G是 k-消去图 .证明了二分图 G=( X,Y) ,且 | X | =| Y|是 k-消去图的充分必要条件是 k| S|≤ r1 + 2 r2 +…+ k( rk+… + rΔ) - ε( S)对所有 S X成立 .并由此给出二分图是 k-消去图的充分度条件 .     

二元图的最佳连通性

本文介绍二元图的最佳连通性问题，给出了有关最佳连通性的若干结果；并就一般情形，给出了二元图最佳连通性问题的解．     

二分图中度条件和k-因子的存在性

本文主要研究了二分图中任意一对距离为２的顶点的度数与ｋ－因子关系,给出了二分图有ｋ因子的若干充分条件,并说明这些条件是最好的可能,从而证明了Ｎｉｓｈｉｍｕｒａ提出的问题对二分图成立。     

On 2-Factors with Prescribed Properties in a Bipartite Graph

Liu and Yan gave the degree condition for a balanced bipartite graph G = （V1, V2; E） to have k vertex-disjoint quadrilaterals containing any given k independent edges e1,……, ek of G, respectively. They also conjectured that for any k independent edges e1,……, ek of G, G has a 2-factor with k cycles C1, C2, ……, Ck with respect to {e1, e2,……, ek} such that k - 1 of them are quadrilaterals. In this paper, we prove this conjecture.     

On the Existence of Disjoint Cycles in a Graph

G is a graph of order at least 3k with . Then G contains k vertex-disjoint cycles. Received: April 23, 1998     

Packing Three Copies of a Tree into a Complete Bipartite Graph

For any tree T of order n, there is a packing of three copies of T into a complete bipartite graph of order at most n + 2. This result is sharp in the sense that n + 2 can not be reduced to n + 1 in general.     

二部定向图的最长通路和圈

Let D be a bipartite oriented graph in which the indegree and outdegree of each vertex are at least k. The result given in this paper is that D contains either a cycle of length at least 4k or a path of length at least 4k + 1. Jackson [1] declared that: if |V(D) |≤4k,D contains a Hamiltonian cycle. Evidently, the result Of this paper implies the result given by Jackson.