3 problems

Odd-girth multicoloring conjecture. If $G$ has odd-girth at least $2k+1$, then $G$ is $(2k+1,k)$-colorable.

McDiarmid–Reed reducibility conjecture. For every hexagonal graph $G$, $\operatorname{core}_{\mathcal{F}_{9,4}\cup\mathcal{F}'_{9,4}}(G)$ is $(9,4)$-colorable.

McDiarmid–Reed's conjecture. Every hexagonal graph is $(9,4)$-colorable.