2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/116409We present a simpler proof of a bound on the number of proper colorings of a graph that was obtained recently by Liu and Murty using Tur'an sieve (in fact, we prove a stronger inequality). We also point out that these results are subsumed in a stronger result due to Lazebnik in 1990.3 pages. Not to be submitted!CombinatoricsNumber Theory05C15text