6 problems

Let $n\in\mathbb{N}$. Clique-minor separating-pair conjecture. There exist non-isomorphic graphs $H$ and $H'$ such that … Here $K_n$ is the complete graph on $n$ vertices. The pape…

Let $\mathcal{F}$ be a minor- and union-closed family of graphs, and let $\cong_{\mathcal{F}}$ denote homomorphism indistinguishability over $\mathcal{F}$. Non-isomorphism conjectu…

Let $\mathcal{F}$ be a family of graphs. Call it homomorphism distinguishing closed if, for every graph $F\notin\mathcal{F}$, there exist graphs $G$ and $H$ such that…

Let $G$ be a connected graph, and let $H$ and $H'$ be graphs. Write $\hom(F,H)$ for the number of homomorphisms from $F$ to $H$. Minor-separating graph-pair conjecture. For every c…

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be minor- and union-closed families of graphs. Write $\cong_{\mathcal{F}_1}\Rightarrow\cong_{\mathcal{F}_2}$ when the first homomorphism ind…

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two distinct minor- and union-closed families of graphs. For a graph family $\mathcal{F}$, write $G\cong_{\mathcal{F}}H$ when $G$ and $H$…