Coloring $(P_5, \text{gem})$-free graphs with $\Delta -1$ colors

TL;DR

This paper proves the Borodin-Kostochka Conjecture for a specific class of graphs that exclude certain induced subgraphs, showing they can be colored with at most  minus one colors.

Contribution

The paper establishes the Borodin-Kostochka Conjecture for $(P_5, ext{gem})$-free graphs, a previously unresolved class of graphs.

Findings

Confirmed the conjecture for $(P_5, ext{gem})$-free graphs

Demonstrated coloring with  - 1 colors is sufficient

Extended understanding of graph coloring in restricted graph classes

Abstract

The Borodin-Kostochka Conjecture states that for a graph $G$, if $\Delta(G) \geq 9$ and $\omega(G) \leq \Delta(G)-1$, then $\chi(G)\leq\Delta(G) -1$. We prove the Borodin-Kostochka Conjecture for $(P_5, \text{gem})$-free graphs, i.e., graphs with no induced $P_5$ and no induced $K_1\vee P_4$.