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

Abstract

AbstractGiven a fixed multigraph H with V(H) = {h1,…, hm}, we say that a graph G is H‐linked if for every choice of m vertices v1, …, vm in G, there exists a subdivision of H in G such that for every i, vi is the branch vertex representing hi. 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