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Let $G = (V,E)$ be a simple, undirected graph. We say that $v\neq w\in V$ lie on a common cycle if there is an integer $n\geq 3$ and an injective graph homomorphism $f: C_n\to V$ such that $v,w\in \...

Let $\mathbb{N}$ denote the set of positive integers, and consider the graph $(\mathbb{N}, E)$ where a set $\{a,b\}$ of two distinct positive integers belongs to $E$ if there is an integer $k>1$ ...

A simple, undirected graph $G=(V,E)$ is said to be vertex-transitive if for all $a,b\in V$ there is a graph isomorphism $\varphi:G\to G$ such that $\varphi(a) = b$. If $G = (\omega, E)$ is vertex-...