Chromatic Uniqueness of \(K_4\)-Homeomorphic Graphs

Abstract

Graphs which are derived from the same graph are called homeomorphic graphs or simply homeomorphs. A \(K_4\)-homeomorph denoted by
\(K_4(a,,c,d,e, f)\) is obtained by subdividing the six paths of a complete graph with four vertices into \(a, b, c,d, e, f\) number of segments, respectively.In this paper, we shall study the chromaticity of \(K_4(a, b,c,d,e, f)\) with exactly two non-adjacent paths of length two. We also give a sufficient and necessary condition for all the graphs in this family to be chromatically
unique.