Non-perfect (P5,C5,K5-e)-Free Graphs are 5-Colorable

Let G be a graph. We use χ ( G ) and ω ( G ) to denote the chromatic number and clique number of G , respectively. A P 5 is a path on 5 vertices, a C 5 is a cycle on 5 vertices, and a K 5 - e is obtained by removing one edge from K 5 . Chudnovsky and Sivaraman showed that χ ( G ) ≤ 2 ω ( G ) - 1 if G is ( P 5 , C 5 ) -free. In this paper, we show that every non-perfect ( P 5 , C 5 , K 5 - e ) -free graph is 5-colorable.