An extremal problem for H‐linked graphs
Alexandr V. Kostochka, Gexin Yu · Journal of Graph Theory · 2005
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 d′ such that every n‐vertex graph with minimum degree at least d′ 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.