6 problems

Covering conjecture for 2-token star graphs. The token graph $F_2(K_{1,n})$ is isomorphic to a covering graph $X^{(\alpha,\omega)}$ of a base graph $X$ on $n-k$ vertices, with comb…

Covering conjecture for odd-star token graphs. The token graph $F_k(K_{1,n})$ is isomorphic to a covering graph $X^{(\alpha,\omega)}$ of a base graph $X$ on

Let $\Gamma=(E,V)$ be a graph, fix a vertex $v$ with non-trivial arborescence, and let $\mathcal V$ be the set of vector fields of $\Gamma$. Let $\Gamma'$ be a uniformly random $k$…

Let $\Gamma$ be a graph and let $\tilde{\Gamma}$ be a $k$-fold cover of $\Gamma$. Let $\mathcal V$ be the set of vector fields of $\Gamma$, let $\operatorname{wt}(\gamma)$ denote t…

Let $\Gamma$ be a directed graph, let $\tilde{\Gamma}$ be a $k$-cover of $\Gamma$, let $v$ be a vertex of $\Gamma$, and let $\tilde{v}$ be a lift of $v$ in $\tilde{\Gamma}$. If the…

Let $\pi\colon \widetilde{\mathbf G}\to\mathbf G=\widetilde{\mathbf G}/\Gamma$ be a maximal Abelian covering of a finite graph, with covering group $\Gamma=\mathbb{Z}^b$, where … i…