4 problems

Let $G$ be a planar graph and let $k$ be an integer. The $k$-recoloring graph $R_k(G)$ has vertices corresponding to proper $k$-colorings of $G$, with edges joining colorings that…

Dvořák–Feghali's conjecture. If $|L(v)|\geq 10$ for every $v\in V(G)$, then

Bartier et al.'s conjecture. If $k\geq d+3$, then

Let $G$ be a planar graph, let $n=|V(G)|$, and let $\mathcal{C}_k(G)$ denote its $k$-recoloring graph. The Bousquet–Bonamy conjecture. The value $k=7$ is enough to ensure that the…