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.