List 2-distance $$\varDelta +3$$ Δ + 3 -coloring of planar graphs without 4,5-cycles

Let $$\chi _2(G)$$ χ 2 ( G ) and $$\chi _2^l(G)$$ χ 2 l ( G ) be the 2-distance chromatic number and list 2-distance chromatic number of a graph G, respectively. Wegner conjectured that for each planar graph G with maximum degree $$\varDelta $$ Δ at least 4, $$\chi _2(G)\le \varDelta +5$$ χ 2 ( G ) ≤ Δ + 5 if $$4\le \varDelta \le 7$$ 4 ≤ Δ ≤ 7 , and $$\chi _2(G)\le \lfloor \frac{3\varDelta }{2}\rfloor +1$$ χ 2 ( G ) ≤ ⌊ 3 Δ 2 ⌋ + 1 if $$\varDelta \ge 8$$ Δ ≥ 8 . Let G be a planar graph without 4,5-cycles. We show that if $$\varDelta \ge 26$$ Δ ≥ 26 , then $$\chi _2^l(G)\le \varDelta +3$$ χ 2 l ( G ) ≤ Δ + 3 . There exist planar graphs G with girth $$g(G)=6$$ g ( G ) = 6 such that $$\chi _2^l(G)=\varDelta +2$$ χ 2 l ( G ) = Δ + 2 for arbitrarily large $$\varDelta $$ Δ . In addition, we also discuss the list L(2, 1)-labeling number of G, and prove that $$\lambda _l(G)\le \varDelta +8$$ λ l ( G ) ≤ Δ + 8 for $$\varDelta \ge 27$$ Δ ≥ 27 .