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arXiv · 2610.40279

The Borodin--Kostochka Conjecture for Two Classes of Diamond-Free Graphs

Alireza Fiujlaali
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Abstract

We prove the Borodin--Kostochka conjecture for $(\mathrm{diamond},L_4)$-free graphs and $(\mathrm{diamond},K_5^+)$-free graphs, where $L_4$ is obtained from $K_4$ by attaching a path of length two to one clique vertex, and $K_5^+$ is obtained from $K_5$ by adding a vertex adjacent to exactly one clique vertex. The proof uses structural properties of maximum cliques in diamond-free vertex-critical graphs with $χ=Δ$.

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Alireza Fiujlaali. 2026-10-29. The Borodin--Kostochka Conjecture for Two Classes of Diamond-Free Graphs. https://arxiv.org/abs/2610.40279

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