A graph minor condition for graphs to be [formula omitted]-linked

Abstract

A graph is called k-linked, if for any 2k distinct vertices x1,x2,…,xk,y1,y2,…,yk, there exist k vertex disjoint paths P1,P2,…,Pk such that Pi connects xi and yi for each 1≤i≤k. Robertson and Seymour showed that every 2k-connected graph having K3k as a minor is k-linked. In 2005, Chen, Gould, Kawarabayashi, Pfender, and Wei proved that every 6-connected graph having K9− as a minor is 3-linked, where Kk−i is the graph obtained from the complete graph with k vertices by deleting exactly i edges. We improve these two results by showing that every 2k-connected graph having the graph obtained from K3k by deleting independent k edges as a minor is k-linked.