Maximizing H ‐Colorings of Connected Graphs with Fixed Minimum Degree

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 K δ , n − δis the unique maximizer. As a corollary, for nonregular H and sufficiently large n the graph K k , n − kis 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 K q ‐colorings (for sufficiently large q and n ) than K δ , n − δ .