Beck's Conjecture for Power Graphs

TL;DR

This paper proves Beck's conjecture for power graphs of finite groups in full generality, providing a comprehensive resolution and clarifying previous ambiguities in the conjecture's formulation.

Contribution

It establishes a complete proof of Beck's conjecture for power graphs of finite groups and refines the statement for a broader class of graphs.

Findings

Confirmed Beck's conjecture for all finite group power graphs

Provided a clearer, more general formulation of the conjecture

Resolved ambiguities in previous conjecture versions

Abstract

Beck's conjecture on coloring of graphs associated to various algebraic objects has generated considerable interest in the community of discrete mathematics and combinatorics since its inception in the year 1988. The version of this conjecture for power-graphs of finite groups has been addressed and partially settled by previous authors. In this paper we answer it in the affirmative in complete generality, and, in effect, we establish a "nicer" statement on a larger class of graphs. We also clear up certain ambiguities present in the way the previous versions of the conjecture were posed.