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The chromatic uniqueness of complete bipartite graphs. (English) Zbl 0752.05031
In the first part of this paper the author looks for the graph in a given class which has the maximum number of induced complete bipartite subgraphs. For graphs with a fixed number of vertices this maximum is achieved by the complete bipartite graph $$K_{n,n}$$ or by $$K_{n,n+1}$$. For graphs with a fixed number of edges the maximum is achieved by $$K_{1,n}$$. The author determines a similar maximum for graphs with a given number of edges and a given maximum degree.
In the second part of the paper the author uses a result from the first half to prove that $$K_{n,m}$$ is chromatically unique when $$n,m\geq 2$$, i.e., that no other graph has the same chromatic polynomial as $$K_{n,m}$$.

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