《中国科学》上“论Ore图的泛连通性”一文的简短证明

给出一种证明Ore图的泛连通性的较简洁证明。     

Bipanconnectivity of faulty hypercubes with minimum degree

In this paper, we consider the conditionally faulty hypercube Q_n with n ? 2 where each vertex of Q_n is incident with at least m fault-free edges, 2 ? m ? n − 1. We shall generalize the limitation m ? 2 in all previous results of edge-bipancyclicity. We also propose a new edge-fault-tolerant bipanconnectivity called k-edge-fault-tolerant bipanconnectivity. A bipartite graph is k-edge-fault-tolerant bipanconnected if G − F remains bipanconnected for any F ⊂ E(G) with ∣F∣ ? k. For every integer m, under the same hypothesis, we show that Q_n is (n − 2)-edge-fault-tolerant edge-bipancyclic and bipanconnected, and the results are optimal with respect to the number of edge faults tolerated. This not only improves some known results on edge-bipancyclicity and bipanconnectivity of hypercubes, but also simplifies the proof.     

A panconnectivity theorem for bipartite graphs

Let $G$ be a simple $m\times n$ bipartite graph with $m\ge n$. We prove that if the minimum degree of $G$ satisfies $\delta \left(G\right)\ge m∕2+1$, then $G$ is bipanconnected: for every pair of vertices $x,y$, and for every appropriate integer $2\le ?\le 2n$, there is an $x,y$-path of length $?$ in $G$.     

Geodesic-pancyclic graphs

A shortest path connecting two vertices u and v is called a u-v geodesic. The distance between u and v in a graph G, denoted by d_G(u,v), is the number of edges in a u-v geodesic. A graph G with n vertices is panconnected if, for each pair of vertices u,v∈V(G) and for each integer k with d_G(u,v)?k?n-1, there is a path of length k in G that connects u and v. A graph G with n vertices is geodesic-pancyclic if, for each pair of vertices u,v∈V(G), every u-v geodesic lies on every cycle of length k satisfying max{2d_G(u,v),3}?k?n. In this paper, we study sufficient conditions of geodesic-pancyclic graphs. In particular, we show that most of the known sufficient conditions of panconnected graphs can be applied to geodesic-pancyclic graphs.