Linking four vertices in graphs of large connectivity

Abstract

One of the most fundamental results in structural graph theory is the “two-paths theorem” that characterizes 2-linkage by planarity. As an extension of the theorem, we consider the following problem for a fixed graph H with four vertices: Given a graph G and an injective map from V(H) to V(G), is there a subdivision of H in G with four branch vertices specified by the map? Hence the case H=2K2 corresponds to the 2-linkage problem. In this paper, for any fixed H with four vertices, we give a structural characterization of 6-connected graphs G with no such subdivision of H. As a corollary, we prove that every 7-connected graph contains a subdivision of K4 with prescribed branch vertices. This generalizes a result of McCarty, Wang and Yu which states that every 7-connected graph is 4-ordered. We also prove that every triangle-free 6-connected graph contains a subdivision of K4 with prescribed branch vertices. This solves a special case of a conjecture of Mader.