Degree sequence conditions for maximally edge-connected graphs and digraphs

In this paper we give simple degree sequence conditions for the equality of edge-connectivity and minimum degree of a (di-)graph. One of the conditions implies results by Bollobás, Goldsmith and White, and Xu. Moreover, we give analogue conditions for bipartite (di-)graphs. © 1997 John Wiley & Sons, Inc. J Graph Theory 26:27–34, 1997     

图的限制性边连通度等于其最小边度的一个充分条件

设G是有限简单无向图．D,g和δ分别表示G的直径,围长和顶点最小度,本文证明,如果D≤g-2,且δ≥3,那么λ＇＝ξ,这里λ＇＝λ＇（G）和ξ＝ξ（G）分别表示G的限制性边连通度和最小边度,它在条件和结论两个方面都改进了已有的研究结果。     

On the restricted connectivity and superconnectivity in graphs with given girth

The restricted connectivity κ^′(G) of a connected graph G is defined as the minimum cardinality of a vertex-cut over all vertex-cuts X such that no vertex u has all its neighbors in X; the superconnectivity κ₁(G) is defined similarly, this time considering only vertices u in G-X, hence κ₁(G)?κ^′(G). The minimum edge-degree of G is ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}, d(u) standing for the degree of a vertex u. In this paper, several sufficient conditions yielding κ₁(G)?ξ(G) are given, improving a previous related result by Fiol et al. [Short paths and connectivity in graphs and digraphs, Ars Combin. 29B (1990) 17-31] and guaranteeing κ₁(G)=κ^′(G)=ξ(G) under some additional constraints.     

Steiner tree packing number and tree connectivity

Let $S$ be a set of at least two vertices in a graph $G$. A subtree $T$ of $G$ is a $S$-Steiner tree if $S?V\left(T\right)$. Two $S$-Steiner trees $T_1$ and $T_2$ are edge-disjoint (resp. internally vertex-disjoint) if $E\left(T_1\right)\cap E\left(T_2\right)=?$ (resp. $E\left(T_1\right)\cap E\left(T_2\right)=?$ and $V\left(T_1\right)\cap V\left(T_2\right)=S$). Let ${\lambda }_G\left(S\right)$ (resp. ${\kappa }_G\left(S\right)$) be the maximum number of edge-disjoint (resp. internally vertex-disjoint) $S$-Steiner trees in $G$, and let ${\lambda }_k\left(G\right)$ (resp. ${\kappa }_k\left(G\right)$) be the minimum ${\lambda }_G\left(S\right)$ (resp. ${\kappa }_G\left(S\right)$) for $S$ ranges over all $k$-subset of $V\left(G\right)$. Kriesell conjectured that if ${\lambda }_G\left(\{x,y\}\right)\ge 2k$ for any $x,y\in S$, then ${\lambda }_G\left(S\right)\ge k$. He proved that the conjecture holds for $\left|S\right|=3,4$. In this paper, we give a short proof of Kriesell’s Conjecture for $\left|S\right|=3,4$, and also show that ${\lambda }_k\left(G\right)\ge ?\frac{1}{k?1}?\frac{k?}{2}??$ (resp. ${\kappa }_k\left(G\right)\ge ?\frac{1}{k?1}?\frac{k?}{2}??$ ) if $\lambda \left(G\right)\ge ?$ (resp. $\kappa \left(G\right)\ge ?$) in $G$, where $k=3,4$. Moreover, we also study the relation between ${\kappa }_k\left(L\left(G\right)\right)$ and ${\lambda }_k\left(G\right)$, where $L\left(G\right)$ is the line graph of $G$.     

On the sizes of graphs and their powers: The undirected case

Let G be an undirected graph and G^r be its r-th power. We study different issues dealing with the number of edges in G and G^r. In particular, we answer the following question: given an integer r≥2 and all connected graphs G of order n such that G^r≠K_n, what is the minimum number of edges that are added when going from G to G^r, and which are the graphs achieving this bound?