A note on coloring vertex-transitive graphs

TL;DR

This paper investigates bounds on the chromatic number of vertex-transitive graphs, proposing a new conjecture relating it to clique number and maximum degree, and proves the Borodin-Kostochka conjecture for graphs with degree at least 13.

Contribution

It introduces a new conjecture on coloring bounds for vertex-transitive graphs and proves the Borodin-Kostochka conjecture for graphs with maximum degree at least 13.

Findings

Proposes a conjecture: hi  eil((5elta + 3)/6)or vertex-transitive graphs.

Proves the Borodin-Kostochka conjecture for vertex-transitive graphs with elta  13.

Provides bounds on hi in terms of lique number nd egree or vertex-transitive graphs.

Abstract

We prove bounds on the chromatic number $\chi$ of a vertex-transitive graph in terms of its clique number $\omega$ and maximum degree $\Delta$. We conjecture that every vertex-transitive graph satisfies $\chi \le \max \left\{\omega, \left\lceil\frac{5\Delta + 3}{6}\right\rceil\right\}$ and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with $\Delta \ge 13$ we prove the Borodin-Kostochka conjecture, i.e., $\chi\le\max\{\omega,\Delta-1\}$.