Beck's Conjecture for Power Graphs

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.