# zbMATH — the first resource for mathematics

Three families of chromatically unique graphs. (English) Zbl 0808.05050
Summary: Let $$P(G)$$ denote the chromatic polynomial of a graph $$G$$. A graph $$G$$ is said to be chromatically unique if $$P (G) = P (H)$$ implies that $$H$$ is isomorphic to $$G$$. We prove that a graph (resp., a bipartite graph) obtained from $$K_{2, 4} \cup P_ s$$ $$(s \geq 3)$$ (resp., $$K_{33} \cup P_ s$$ $$(s \geq 7)$$) by identifying the end vertices of the part $$P_ s$$ with any two vertices of the complete bipartite graph $$K_{2, 4}$$ (resp., $$K_{3, 3})$$ is chromatically unique. We also show that a bipartite graph obtained from $$(K_{3, 3}-e) \cup P_ s$$ $$(s \geq 5)$$ where $$e$$ is an edge of $$K_{3, 3,}$$, by identifying the end vertices of the path $$P_ s$$ with two nonadjacent vertices of $$K_{3, 3}-e$$ is chromatically unique.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
chromatically unique graphs; chromatic polynomial