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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
##### Keywords:
uniquely colorable; bipartite graphs; threshold graphs
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