Every $3$-connected $\{K_{1,3},\Gamma_3\}$-free graph is Hamilton-connected

TL;DR

This paper proves that all 3-connected graphs avoiding certain subgraphs are Hamilton-connected, using a new closure technique and structural analysis, with some steps verified by computer.

Contribution

It establishes the Hamilton-connectedness of 3-connected {K_{1,3}, Γ_3}-free graphs, resolving a key open case with a novel closure method.

Findings

All such graphs are Hamilton-connected.

The proof introduces a new closure technique.

Computer-assisted analysis was employed for complex steps.

Abstract

We show that every $3$-connected $\{K_{1,3},\Gamma_3\}$-free graph is Hamilton-connected, where $\Gamma_3$ 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. Keywords: Hamilton-connected; closure; forbidden subgraph; claw-free; $\Gamma_3$-free