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arXiv · 2410.18309

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

Adam Kabela·Zdeněk Ryjáček·Mária Skyvová·Petr Vrána
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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

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Adam Kabela, Zdeněk Ryjáček, Mária Skyvová, Petr Vrána. 2024-10-23. Every $3$-connected $\{K_{1,3},\Gamma_3\}$-free graph is Hamilton-connected. https://arxiv.org/abs/2410.18309

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