On H ‐immersions

For a fixed multigraph H , possibly containing loops, with V ( H ) = { h 1 ,…, h k }, we say a graph G is H ‐linked if for every choice of k vertices v 1 ,…, v k in G , there exists a subdivision of H in G such that v i represents h i (for all i ). An H ‐immersion in G is similar except that the paths in G , playing the role of the edges of H , are only required to be edge disjoint. In this article, we extend the notion of an H ‐linked graph by determining minimum degree conditions for a graph G to contain an H ‐immersion with a bounded number of vertex repetitions on any choice of k vertices. In particular, we extend results found in [2,3,5]. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 245–254, 2008