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

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 $χ=Δ$.