Short proofs for two theorems of Chien, Hell and Zhu

Abstract

In (J Graph Theory 33 (2000), 14–24), Hell and Zhu proved that if a series–parallel graph G has girth at least 2⌊(3k−1)/2⌋, then χc(G)≤4k/(2k−1). In (J Graph Theory 33 (2000), 185–198), Chien and Zhu proved that the girth condition given in (J Graph Theory 33 (2000), 14–24) is sharp. Short proofs of both results are given in this note. © 2010 Wiley Periodicals, Inc. J Graph 66: 83-88, 2010 Theory © 2011 Wiley Periodicals, Inc.