List-coloring the Squares of Planar Graphs without 4-Cycles and 5-Cycles

Abstract

Let $G$ be a planar graph without 4-cycles and 5-cycles and with maximum degree $Δ\ge 32$. We prove that $χ_{\ell}(G^2)\le Δ+3$. For arbitrarily large maximum degree $Δ$, there exist planar graphs $G_Δ$ of girth 6 with $χ(G_Δ^2)=Δ+2$. Thus, our bound is within 1 of being optimal. Further, our bound comes from coloring greedily in a good order, so the bound immediately extends to online list-coloring. In addition, we prove bounds for $L(p,q)$-labeling. Specifically, $λ_{2,1}(G)\le Δ+8$ and, more generally, $λ_{p,q}(G)\le (2q-1)Δ+6p-2q-2$, for positive integers $p$ and $q$ with $p\ge q$. Again, these bounds come from a greedy coloring, so they immediately extend to the list-coloring and online list-coloring variants of this problem.