Maximizing H‐Colorings of Connected Graphs with Fixed Minimum Degree |
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Authors: | John Engbers |
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Affiliation: | DEPARTMENT OF MATHEMATICS, STATISTICS AND COMPUTER SCIENCE, MARQUETTE UNIVERSITY, MILWAUKEE, WISCONSIN |
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Abstract: | For graphs G and H , an H‐coloring of G is a map from the vertices of G to the vertices of H that preserves edge adjacency. We consider the following extremal enumerative question: for a given H , which connected n‐vertex graph with minimum degree δ maximizes the number of H‐colorings? We show that for nonregular H and sufficiently large n , the complete bipartite graph is the unique maximizer. As a corollary, for nonregular H and sufficiently large n the graph is the unique k‐connected graph that maximizes the number of H‐colorings among all k‐connected graphs. Finally, we show that this conclusion does not hold for all regular H by exhibiting a connected n‐vertex graph with minimum degree δ that has more ‐colorings (for sufficiently large q and n ) than . |
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Keywords: | graph homomorphism H‐coloring proper coloring minimum degree k‐connected |
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