Homomorphism counting for immersion-closed classes is not isomorphism

Abstract

Lov\'{a}sz proved that two graphs $G$ and $H$ are isomorphic if $\hom(K,G) = \hom(K,H)$ for all graphs $K$, where $\hom(G_1,G_2)$ denotes the number of homomorphisms from $G_1$ to $G_2$. Dvo\v{r}\'{a}k showed that it suffices to count homomorphisms from all $2$-degenerate graphs $K$. On the other hand, for several interesting graph classes $\mathcal{M}$, it has been shown that there exist non-isomorphic graphs $G$ and $H$ such that $\hom(K,G)=\hom(K,H)$ for all $K\in \mathcal{M}$. Most such classes are minor-closed and Roberson conjectured that every proper minor-closed graph class $\mathcal{M}$ has the property that there exist non-isomorphic graphs that are indistinguishable by homomorphism counts from $\mathcal{M}$. There has been an effort to prove Roberson's conjecture as it is believed that minor-closed classes play a special role in demarcating the graph classes that satisfy this property. We show that this special role, if it exists, must be shared, by proving an analogue of Roberson's conjecture for a rich family of non-minor-closed classes. Namely, we prove that for any proper immersion-closed graph class $\mathcal{M}$, there exist non-isomorphic graphs $G$ and $H$ such that $\hom(K,G) = \hom(K,H)$ for all $K \in \mathcal{M}$. This extends a result of Roberson on homomorphism indistinguishability over bounded degree graphs, and cannot be extended in the natural way by replacing immersions with topological minors due to a result of Neuen and Seppelt. Our main result is obtained as a consequence of a colouring result which may be of independent interest: Every graph with a $\oddbound$-oddomorphism admits a $K_{t}$-immersion.