Graphs with large generalized 3-edge-connectivity

Details Graphs with large generalized 3-edge-connectivity Graphs with large generalized 3-edge-connectivity

2012-01-18

arxiv.org/abs/1201.3699v1

Mathematics

Combinatorics

9 pages

Scientific paper

Let $G$ be a graph, and $S$ a set of vertices of $G$. Denote by $\lambda (S)$ the maximum number $\ell$ of pairwise edge-disjoint trees $T_1, T_2, ..., T_\ell$ in $G$ such that $V(T_i)\supseteq S$ for every $1 \leq i \leq \ell$. Then the generalized $k$-edge-connectivity $\lambda_k(G)$ of $G$ is defined as $\lambda_k(G) = min{\lambda (S)|S \subseteq V(G) and |S| = k}$. Thus $\lambda_2(G) = \lambda (G)$. In this paper, sharp upper and lower bounds of $\lambda_3(G)$ are given for a connected graph $G$ of order $n$, that is, $1 \leq \lambda_3(G) \leq n - 2$. Graphs of order $n$ such that $\lambda_3(G) = n - 2, n - 3$ are characterized, respectively.

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