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  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.
MSU Digital Commons Citation
Cutler, Jonathan and Radcliffe, A. J., "Extremal Graphs for Homomorphisms II" (2014). Department of Mathematics Facuty Scholarship and Creative Works. 75.