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The chromaticity of complete bipartite graphs with at most one edge deleted. (English) Zbl 0712.05027
The chromatic polynomial of a graph G is denoted by P(G,$$\lambda$$). Two graphs are chromatically equivalent if they have the same chromatic polynomials. A graph G is said to be chromatically unique if every graph H with $$P(G,\lambda)=P(H,\lambda)$$ is isomorphic with G.
The paper concerns with the characterization of chromatically unique graphs. All complete bipartitie graphs K(p,q), $$2\leq p\leq q,$$ are chromatically unique. All graphs obtained from a complete bipartite graph K(p,q) by deleting one edge, where $$3\leq p\leq q$$, are also chromatically unique. These results prove a conjecture of Salzberg, López, and Giudici.
Open questions concern the chromaticity of $$K(p,q)-\{e_ 1,e_ 2\}$$ with $$4\leq p\leq q$$ $$(e_ 1,e_ 2$$ are two distinct edges) and of $$K(p,q)+e$$ with $$2\leq p\leq q$$ (e is a new edge joining two vertices in a partite set).
Reviewer: U.Baumann

##### MSC:
 05C15 Coloring of graphs and hypergraphs
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