10 problems

General Cartesian-product conjecture. Every precoloring of at most $k+l+1$ edges of $G \square H$ is extendable to a proper $(\chi'(G)+\chi'(H))$-edge-coloring of $G \square H$.

Casselgren, Markström and Pham's conjecture. If $n$ and $d$ are positive integers, and $\varphi$ is a proper edge-precoloring of $K_{n,n}^d$ with at most $nd-1$ precolored edges, t…

Balanced complete-bipartite Cartesian-product conjecture. If any precoloring of at most $k < r$ edges of $G$ can be extended to a proper $\chi'(G)$-edge-coloring of $G$, then any p…

Casselgren, Petros and Fufa's conjecture. If every precoloring of at most $k < \chi'(G)$ edges of $G$ can be extended to a proper $\chi'(G)$-edge-coloring, then every precoloring o…

Cartesian-product precoloring extension conjecture. This precoloring is extendable to a proper edge coloring.

Casselgren–Markström–Pham conjecture. Any precolored distance-2 matching in $Q_d$ can be extended to a proper $d$-edge coloring.

Generic precoloring-extension conjecture. For every positive even integer $\ell$, there exists an integer $d$ such that, if no 4-cycle in $G-U$ separates a face of $S$ from the out…

Edwards et al.'s conjecture. For any graph $G$, any precoloring of a distance-$2$ matching can be extended to a proper edge coloring of $G$ using colors from $[\Delta(G)+\mu(G)]$.

Albertson–Moore conjecture. Any precoloring of a distance-$3$ matching can be extended to a proper edge coloring of $G$ using colors from $[\Delta(G)+1]$.

Let $G$ be a planar graph with two precolored vertices that are sufficiently far apart, and let the remaining vertices have lists of size $5$. A precoloring of the two vertices is…