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On 3-colorings of partite p-threshold graphs. (English) Zbl 0662.05024
Let P(G,\(\lambda)\) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent if \(P(G,\lambda)=P(H,\lambda)\). G is chromatically unique if it is chromatically equivalent to no other graph. The author defines a bipartite p-threshold graph which is a generalization of a threshold graph. Loosely speaking, in a bipartite p-threshold graph any two vertices which are not comparable belong to different partite sets. He examines properties of 3-colorings of complete bipartite graphs which are extremely in the class of all bipartite p-threshold graphs that are uniquely 2-colorable. As a consequence it is shown that the complete bipartite graphs \(K_{p,p+r}\) are chromatically unique if \(p\geq 2\) and \(0\leq r<2\sqrt{p+1}\).
Reviewer: D.S.Archdeacon

MSC:
05C15 Coloring of graphs and hypergraphs
05C99 Graph theory
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