Extremal H-colorings of trees and 2-connected graphs

Abstract For graphs G and H , an H-coloring of G is an adjacency preserving map from the vertices of G to the vertices of H . H -colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of H -colorings. In this paper, we prove several results in this area. First, we find a class of graphs H with the property that for each H ∈ H , the n -vertex tree that minimizes the number of H -colorings is the path P n . We then present a new proof of a theorem of Sidorenko, valid for large n , that for every H the star K 1 , n − 1 is the n -vertex tree that maximizes the number of H -colorings. Our proof uses a stability technique which we also use to show that for any non-regular H (and certain regular H ) the complete bipartite graph K 2 , n − 2 maximizes the number of H -colorings of n -vertex 2-connected graphs. Finally, we show that the cycle C n has the most proper q -colorings among all n -vertex 2-connected graphs.