On chromatic equivalence of a pair of K4-homeomorphs

Abstract

Let $P(G, \lambda)$ be the chromatic polynomial of a graph $G$. Two graphs $G$ and $H$ are said to be chromatically equivalent, denoted $G∼H$, if $PP(G, \lambda)=P(H, \lambda)$. We write $[G] = \{H| H \sim G\}$. If $[G] = \{G\}$, then $G$ is said to be chromatically unique. In this paper, we discuss a chromatically equivalent pair of graphs in one family of $K_{4}$-homeomorphs, $K_{4}(1,2,8,d,e,f)$. The obtained result can be extended in the study of chromatic equivalence classes of $K_{4}(1,2,8,d,e,f)$ and chromatic uniqueness of $K_{4}$-homeomorphs with girth $11$.