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

Let G be a planar graph without 4-cycles and 5-cycles and with maximum degree ∆ ≥ 32. We prove that χ`(G ) ≤ ∆ + 3. For arbitrarily large maximum degree ∆, there exist planar graphs G∆ of girth 6 with χ(G 2 ∆) = ∆ + 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, λ2,1(G) ≤ ∆ + 8 and, more generally, λp,q(G) ≤ (2q − 1)∆ + 6p − 2q − 2, for positive integers p and q with p ≥ 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.