Graph operations and upper bounds on graph homomorphism counts

Abstract

We construct a family of countexamples to a conjecture of Galvin [5], which stated that for any n‐vertex, d‐regular graph G and any graph H (possibly with loops), homfalse(G,Hfalse)≤max{hom(Kd,d,H)n2d,hom(Kd+1,H)nd+1},where hom(G,H) is the number of homomorphisms from G to H. By exploiting properties of the graph tensor product and graph exponentiation, we also find new infinite families of H for which the bound stated above on hom(G,H) holds for all n‐vertex, d‐regular G. In particular, we show that if HWR is the complete looped path on three vertices, also known as the Widom–Rowlinson graph, then homfalse(G,H WR false)≤hom(Kd+1,H WR )nd+1for all n‐vertex, d‐regular G. This verifies a conjecture of Galvin.

Extremal Regular Graphs: Independent Sets and Graph Homomorphisms

“…It was conjectured [32] that one of K d,d and K d+1 always maximizes hom(G, H) 1/v(G) for every H. However, Sernau [52] showed that this is false (a similar construction was independently found by Pat Devlin).…”

“…The following special case, proved in [10] (prior to [52]), provides some a nice family of examples.…”

“…The results have been partially extended to graph homomorphisms, though many intriguing open problems remain. We also discuss some recent work on the subject done by Luke Sernau [52] as an undergraduate student at Notre Dame.…”

“…Sernau [52] extended the class of H for which Theorem 4.3 holds by observing that this class is closed under taking tensor products. See Section 5.2.…”

“…Sernau [52] observed that, for any d, if H = H 1 and H = H 2 both have the property that G = K d,d maximizes hom(G, H) 1/v(G) over all d-regular graphs G, then H = H 1 × H 2 has the same property by (4). In other words, the set of H such that G = K d,d is the maximizer in Problem 2.5 is closed under tensor products.…”