Every 3-connected {K1,3,Γ3}-free graph is Hamilton-connected

Abstract

We show that every 3-connected {K1,3, Γ3}-free graph is Hamilton-connected, where Γ3 is the graph obtained by joining two vertex-disjoint triangles with a path of length 3. This resolves one of the two last open cases in the characterization of pairs of connected forbidden subgraphs implying Hamilton-connectedness. The proof is based on a new closure technique, developed in a previous paper, and on a structural analysis of small subgraphs, cycles and paths in line graphs of multigraphs. The most technical steps of the analysis are computer-assisted.