Minimum degree conditions for H-linked graphs

For a fixed multigraph H with vertices w1, . . . , wm, a graph G is H-linked if for every choice of vertices v1, . . . , vm in G, there exists a subdivision of H in G such that vi is the branch vertex representing wi (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs. Given a connected multigraph H with k edges and minimum degree at least two and n 7.5k, we determine the least integer d such that every n-vertex simple graph with minimum degree at least d is H-linked. This value D(H, n) appears to equal the least integer d ′ such that every n-vertex graph with minimum degree at least d ′ is b(H)-connected, where b(H) is the maximum number of edges in a bipartite subgraph of H. © 2007 Elsevier B.V. All rights reserved.