Dingjun Lou

M -alternating paths in n -extendable bipartite graphs

Let G be a bipartite graph with bipartition (X,Y) which has a perfect matching. It is proved that G is n-extendable if and only if for any perfect matching M of G and for each pair of vertices x in X and y in Y there are n internally disjoint M-alternating paths connecting x and y. Furthermore, these n paths start and end with edges in E(G)\M. This theorem is then generalized.

The Chvátal-Erdos condition for cycles in triangle-free graphs

Abstract It is proved that if G is a triangle-free graph with v vertices whose independence number does not exceed its connectivity then G has cycles of every length n for 4 ⩽ n ⩽ v ( G ) unless G = K v /2, v /2 or G is a 5-cycle. This was conjectured by Amar, Fournier and Germa.

On the structure of minimally n -extendable bipartite graphs

Abstract It is proved that, in a minimal n-extendable bipartite graph, the subgraph induced by the edges both ends of which have degree at least n + 2 is a forest. As a consequence, every minimal n-extendable bipartite graph has at least 2n + 2 vertices of degree n + 1. This result is sharp. Some other structural results on minimally n-extendable bipartite graphs are also given.

Characterizing defect n-extendable bipartite graphs with different connectivities

A near perfect matching is a matching saturating all but one vertex in a graph. If G is a connected graph and any n independent edges in G are contained in a near perfect matching where n is a positive integer and n= =2 and @k(G)>=n, respectively. Some properties for defect n-extendable bipartite graphs with different connectivities are also given.

A note on internally disjoint alternating paths in bipartite graphs

Abstract Let G be a balanced bipartite graph with partite sets X and Y, which has a perfect matching, and let x ∈ X and y ∈ Y . Let k be a positive integer. Then we prove that if G has k internally disjoint alternating paths between x and y with respect to some perfect matching, then G has k internally disjoint alternating paths between x and y with respect to every perfect matching.

Vertex-disjoint cycles containing specified vertices in a bipartite graph: VERTEX-DISJOINT CYCLES IN BIPARTITE GRAPHS

A minimum degree condition is given for a bipartite graph to contain a 2-factor each component of which contains a previously specified vertex. 2004 Wiley Periodicals, Inc. J Graph Theory 46: 145–166, 2004

Characterizing 2k-critical graphs and n-extendable graphs

Abstract Let G be a graph with even order. Let M be a matching in G and x 1 , x 2 , … , x 2 r , be the M -unsaturated vertices in G . Then G has a perfect matching if and only if there are r independent M -augmenting paths joining the 2 r vertices in pairs. Let G be a graph with a perfect matching M . It is proved that G is 2 k -critical if and only if for any 2 k vertices u 1 , u 2 , … , u 2 k in G , there are k independent M -alternating paths P 1 , P 2 , … , P k joining the 2 k vertices in pairs such that P 1 , P 2 , … , P k start and end with edges in M . It is also proved that G is n -extendable if and only if, for each r with 0 ⩽ r ⩽ n and each F ⊂ M with | F | = r , and for any n - r pairs of M -alternating paths x i x i ′ y i ′ y i with x i x i ′ , y i y i ′ ∈ M ( 1 ⩽ i ⩽ n - r ) in G - V ( F ) , there exist independent M -alternating paths P 1 , P 2 , … , P k in G - V ( F ) joining the vertices in Z = { x 1 , y 1 , … , x n - r , y n - r } ⧹ { x 1 ′ , y 1 ′ , … , x n - r ′ , y n - r ′ } , where | Z | = 2 k , which start and end with edges in E ( G ) ⧹ M .

Partitioning regular graphs into equicardinal linear forests

Abstract It is a well-known fact that the linear arboricity of a k-regular graph is ⌈ (k + 1) 2 ⌉ for k = 3,4. In this paper, we prove that if the number of edges of a k-regular graph is divisible by ⌈ (k + 1) 2 ⌉ , then its edge set can be partitioned into ⌈ (k + 1) 2 ⌉ linear forests, all of which have the same number of edges (k = 3,4).

Characterization of graphs with infinite cyclic edge connectivity

In this paper, we characterize the graphs with infinite cyclic edge connectivity. Then we design an efficient algorithm to determine whether a graph has finite cyclic edge connectivity or infinite cyclic edge connectivity.

Graphs with no M-alternating path between two vertices

Let G be a graph with a perfect matching M. In this paper, we prove two theorems to characterize the graph G in which there is no M-alternating path between two vertices x and y in G.