9 problems

Let $G=(V,E)$ be a graph, let $\tau(G)$ denote its covering number, and let $L(G)$ be the Laplacian matrix of $G$. For $1\leq i\leq |V|$, write $\lambda_i(L(G))$ for the $i$-th lar…

Asymptotic directed-cycle packing and covering conjecture. For all sufficiently large $n$, every $n$-vertex directed graph $D$ satisfies

The stronger-than relation conjecture. If $A$ has no semi-edges, then

Let $K_t(d)$ be the complete $t$-partite graph with each part of size $d$. Davoodi–Javadi–Omoomi's conjecture. There exists a function $f$ and a constant $c>0$ such that, for every…

Let $\Gamma$ be an infinite Cayley graph that is not quasi-isometric to $\mathbf Z$. A graph $\Delta$ is a cover of $\Gamma$ if there is a covering map from $\Gamma$ to $\Delta$; w…

Let $\Omega$ be a finite connected graph, let $\Gamma$ be a random $n$-covering of $\Omega$, and let $\lambda_A(\Gamma)$ and $\lambda_M(\Gamma)$ denote the largest absolute values…

Let $K_n$ be a complete graph whose edges are coloured with $r$ colours. A cycle covering is a collection of $r$ monochromatic cycles whose union of vertex sets contains all vertic…

Let $K_n$ be a complete graph whose edges are coloured with $r$ colours. A vertex-disjoint monochromatic cycle packing is a collection of $r$ vertex-disjoint monochromatic cycles.…

Hamilton covering conjecture. For any $p=\omega(\ln n)/n$, the random graph $G(n,p)$ a.a.s. admits a covering of its edges with at most