Counterexamples to the Strong Roberson Conjecture

Abstract

We refute the Strong Roberson Conjecture, which asserts that adding any graph outside a class closed under minors and disjoint unions strictly increases the distinguishing power of homomorphism counts from that class. More precisely, we construct connected graphs $H$ for which counts from graphs excluding $H$ as a minor determine the number of homomorphisms from $H$ to any target graph. We also refute the analogous conjecture with immersions in place of minors. We give explicit infinite families of excluded graphs, including cubic bipartite graphs that yield counterexamples for both relations. The proof introduces a method for deriving exact homomorphism count dependence from modular equivalences. We obtain these equivalences for infinitely many primes using prime-order automorphisms of graphs that exclude their orbit quotients as minors or immersions.