On the chromatic number of the power graph of a finite group

The power graph ΓGΓG of a finite group GG is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. We investigate the chromatic number χ(ΓG)χ(ΓG) of ΓGΓG. A characterization of χ(ΓG)χ(ΓG) is presented, and a conjecture in Mirzargar et al. (2012) is disproved. Moreover, we classify all finite groups whose power graphs are uniquely colorable, split or unicyclic.