Extremal H-Colorings of Graphs with Fixed Minimum Degree

For graphs G and H, a homomorphism from G to H, or H-coloring of G, is a map from the vertices of G to the vertices of H that preserves adjacency. When H is composed of an edge with one looped endvertex, an H-coloring of G corresponds to an independent set in G. Galvin showed that, for sufficiently large n, the complete bipartite graph Kδ,n-δ is the n-vertex graph with minimum degree δ that has the largest number of independent sets. In this paper, we begin the project of generalizing this result to arbitrary H. Writing hom(G, H) for the number of H-colorings of G, we show that for fixed H and δ = 1 or δ = 2, 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 G with minimum degree δ (for sufficiently large n). We also provide examples of H for which the maximum is achieved by hom(Kδ+1, H)n⁄δ+1 and other H for which the maximum is achieved by hom(Kδ,δ,H)n⁄2δ. For δ ≥ 3 (and sufficiently large n), we provide a infinite family of H for which hom(G, H) ≤ hom (Kδ,n-δ, H) for any n-vertex G with minimum degree δ. The results generalize to weighted H-colorings

Extremal Graphs for Homomorphisms II

Extremal problems for graph homomorphisms have recently become a topic of much research. Let hom(G,H) denote the number of homomorphisms from G to H. A natural set of problems arises when we fix an image graph H and determine which graph(s) G on n vertices and m edges maximize hom(G,H). We prove that if H is loop-threshold, then, for every n and m, there is a threshold graph G with n vertices and m edges that maximizes hom(G,H). Similarly, we show that loop-quasi-threshold image graphs have quasi-threshold extremal graphs. In the case H=P3o, the path on three vertices in which every vertex in looped, the authors [5] determined a set of five graphs, one of which must be extremal for hom(G,P3o). Also in this article, using similar techniques, we determine a set of extremal graphs for the fox, a graph formed by deleting the loop on one of the end-vertices of P3o. The fox is the unique connected loop-threshold image graph on at most three vertices for which the extremal problem was not previously solved