An extremal problem for H -linked graphs

TLDR: The notion of H-linked graphs was introduced in this paper, where a graph G is a fixed multigraph with vertices w1,…,wm, wm in G such that υi is the branch vertex representing wi (for all i) in G. This generalizes the notions of k-linked, k-connected and k-ordered graphs.

Abstract: We introduce the notion of H-linked graphs, where H is a fixed multigraph with vertices w1,…,wm. A graph G is H-linked if for every choice of vertices υ1,…, υm in G, there exists a subdivision of H in G such that υi is the branch vertex representing wi (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs. Given k and n ≥ 5k+6, we determine the least integer d such that, for every loopless graph H with k edges and minimum degree at least two, every n-vertex graph with minimum degree at least d is H-linked. This value D1(k,n) appears to equal the least integer d2 such that every n-vertex graph with minimum degree at least d2 is k-connected. On the way to the proof, we extend a theorem by Kierstead et al. on the least integer d~ such that every n-vertex graph with minimum degree at least d~ is k-ordered. © 2005 Wiley Periodicals, Inc.