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 Kδ,n−δ is the unique maximizer. As a corollary, for nonregular H and sufficiently large n the graph Kk,n−k 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 Kq‐colorings (for sufficiently large q and n) than Kδ,n−δ.