A Family of Chromatically Unique $6$-bridge Graphs

Abstract

Let $P(G,\lambda )$ denote the chromatic polynomial of a graph $G$. Two graphs $G$ and $H$ are chromatically equivalent, written $G\sim H$, if $P(G,\lambda )=P(H,\lambda )$. A graph $G$ is chromatically unique, written $x$-unique, if for any graph $H$, $G\sim H$ implies that $G$ is isomorphic with $H$. In this paper, we prove that the graph $\theta (a_1,a_2,\dots ,a_6)$ is $x$-unique for exactly two distinct values of $a_1,a_2,\dots ,a_6$.