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.