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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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