A closure for Hamilton-connectedness in {K1,3,Γ3}-free graphs

Abstract

We introduce a closure technique for Hamilton-connectedness of {K(1,3),Gamma(3)}-free graphs, where Gamma(3) is the graph obtained by joining two vertex-disjoint triangles with a path of length 3. The closure turns a claw-free graph into a line graph of a multigraph while preserving its (non)-Hamilton-connectedness. The most technical parts of the proof are computer-assisted.The main application of the closure is given in a subsequent paper showing that every 3-connected {K(1,3),Gamma(3)}-free graph is Hamilton-connected, thus resolving one of the two last open cases in the characterization of pairs of connected forbidden subgraphs implying Hamilton-connectedness.