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On Sufficient Degree Conditions for a Graph to be $k$-linked

Published online by Cambridge University Press:  31 July 2006

KEN-ICHI KAWARABAYASHI
Affiliation:
Graduate School of Information Sciences (GSIS), Tohoku University, Aramaki aza Aoba 09, Aoba-ku Sendai, Miyagi, 980-8579, Japan (e-mail: k_keniti@dais.is.tohoku.ac.jp)
ALEXANDR KOSTOCHKA
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA and Institute of Mathematics, Novosibirsk 630090, Russia (e-mail: kostochk@math.uiuc.edu)
GEXIN YU
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: gexinyu@uiuc.edu)

Abstract

A graph is $k$-linked if for every list of $2k$ vertices $\{s_1,{\ldots}\,s_k, t_1,{\ldots}\,t_k\}$, there exist internally disjoint paths $P_1,{\ldots}\, P_k$ such that each $P_i$ is an $s_i,t_i$-path. We consider degree conditions and connectivity conditions sufficient to force a graph to be $k$-linked.

Let $D(n,k)$ be the minimum positive integer $d$ such that every $n$-vertex graph with minimum degree at least $d$ is $k$-linked and let $R(n,k)$ be the minimum positive integer $r$ such that every $n$-vertex graph in which the sum of degrees of each pair of non-adjacent vertices is at least $r$ is $k$-linked. The main result of the paper is finding the exact values of $D(n,k)$ and $R(n,k)$ for every $n$ and $k$.

Thomas and Wollan [14] used the bound $D(n,k)\leq (n+3k)/2-2$ to give sufficient conditions for a graph to be $k$-linked in terms of connectivity. Our bound allows us to modify the Thomas–Wollan proof slightly to show that every $2k$-connected graph with average degree at least $12k$ is $k$-linked.

Type
Paper
Copyright
2005 Cambridge University Press

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