Malaysian Journal of Mathematical Sciences, September 2015, Vol. 9, No. 3
Chromaticity of a Family of $K_4$-Homeomorphs with Girth 9, II
N. S. A. Karim, R. Hasni, and G. C. Lau
Corresponding Email: hroslan@umt.edu.my
Received date: -
Accepted date: -
Abstract:
For a graph $G$, let $P(G, \lambda)$ denote the chromatic polynomial of $G$. Two graphs $G$ and $H$ are chromatically equivalent (or simply $\chi$-equivalent), denoted by $G \sim H $, if $P(G, \lambda) = P(H, \lambda)$. A graph $G$ is chromatically unique (or simply $\chi$-unique) if for any graph $H$ such as $H \sim G $, we have $H \cong G$, i.e, $H$ is isomorphic to $G$. A $K_4$-homeomorph is a subdivision of the complete graph $K_4$. In this paper, we investigate the chromaticity of one family of $K_4$-homeomorphs which has girth 9, and give sufficient and necessary condition for the graph in the family to be chromatically unique.
Keywords: Chromatic polynomial, Chromatically unique, $K_4$-homeomorphs
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