Some Problems Involving H-colorings of Graphs

by John Alan Engbers For graphs G and H, an H-coloring of G, or homomorphism from G to H, is an edge-preserving map from the vertices of G to the vertices of H. H-colorings generalize such graph theory notions as proper colorings and independent sets. In this dissertation, we consider four questions involving H-colorings of graphs. Recently, Galvin [27] showed that the maximum number of independent sets in a an n vertex minimum degree δ graph occurs (for sufficiently large n) whenG = Kδ,n−δ. First, we show this result holds for level sets: for all triples (n, δ, t) with δ ≤ 3 and t ≥ 3, no n-vertex graph with minimum degree δ admits more independent sets of size t than Kδ,n−δ, and we obtain the same conclusion for δ > 3 and t ≥ 2δ + 1. Second, we begin the project of generalizing Galvin’s result to arbitrary H. Writing hom(G,H) for the number of H-colorings of G, we show that for δ = 1 and δ = 2 and fixed H, hom(G,H) ≤ max{hom(Kδ+1, H) n δ+1 , hom(Kδ,δ, H) n 2δ , hom(Kδ,n−δ, H)} for any n vertex minimum degree δ graph G (for sufficiently large n). For δ ≥ 3 (and sufficiently large n), we provide a class of H for which hom(G,H) ≤ hom(Kδ,n−δ, H) for any G in this family.