Ore‐type degree conditions for a graph to be H ‐linked

Given a fixed multigraph H with V ( H ) = { h 1 ,…, h m }, we say that a graph G is H ‐linked if for every choice of m vertices v 1 , …, v m in G , there exists a subdivision of H in G such that for every i , v i is the branch vertex representing h i . This generalizes the notion of k ‐linked graphs (as well as some other notions). For a family ${\cal H}$ of graphs, a graph G is ${\cal H}$ ‐linked if G is H ‐linked for every $H\in {\cal H}$ . In this article, we estimate the minimum integer r  =  r ( n , k , d ) such that each n ‐vertex graph with $\sigma_{2}(G)\ge {r}$ is ${\cal H}$ ‐linked, where ${\cal H}$ is the family of simple graphs with k edges and minimum degree at least $d \ge 2$ . © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 14–26, 2008