Minimum degree conditions for [formula omitted]-linked graphs

Abstract

For a fixed multigraph H with vertices w 1 , … , w m , a graph G is H- linked if for every choice of vertices v 1 , … , v m in G, there exists a subdivision of H in G such that v i is the branch vertex representing w i (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 ⩾ 7.5 k , 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.