Every $3$-connected $\{K_{1,3},Γ_3\}$-free graph is Hamilton-connected

Abstract

We show that every $3$-connected $\{K_{1,3},Γ_3\}$-free graph is Hamilton-connected, where $Γ_3$ 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; $Γ_3$-free

Figures (5)

• Figure 1: The Wagner graph $W$ and the graph $W^+$
• Figure 2: The graph $\Gamma_3$ and its three preimages
• Figure 3: The multitriangle, the diamond $D$, and the graphs $D^1$ and $D^2$.
• Figure 4: The graph $K_{2,r}$, a multigraph from $\mathbb{K}_{2,r}^M$ and a multigraph from $\mathbb{K}_{rP_4}^M$.
• Figure 5: The multigraphs $W_1$ and $W_2$, and the graph $W_3$.