The Condition of Beineke and Harary on Edge-Disjoint Paths Some of Which are Openly Disjoint

Abstract

A pair $(t,s)$ of nonnegative integers is said to be a connectivity pair for distinct vertices $x$ and $y$ of a graph $G$ if it satisfies the following conditions which were introduced by Beineke and Harary: \begin{enumerate} \item[(1)] For any subset $T\subseteq V(G)-\{x,y\}$ and any subset $S\subseteq E(G)$ with $|T|\leqq t$, $|S|\leqq s$ and $|T|+|S|<t+s$, $G-(T\cup S)$ still contains an $x$-$y$ path. \item[(2)] There exist a subset $T'\subseteq V(G)-\{x,y\}$ and a subset $S'\subseteq E(G)$ with $|T'|=t$ and $|S'|=s$ such that $G-(T'\cup S')$ contains no $x$-$y$ path. \end{enumerate} Let $q$, $r$, $s$ and $t$ be integers with $t\geqq 0$ and $s\geqq 1$ such that $t+s=q(t+1)+r$, $1\leqq r\leqq t+1$, and let $x$ and $y$ be distinct vertices of a graph $G$. It is shown that if $q+r>t$ and if $(t,s)$ is a connectivity pair for $x$ and $y$, then $G$ contains $t+s$ edge-disjoint $x$-$y$ paths $t+1$ of which are openly disjoint.