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

Let $$\chi _2(G)$$χ2(G) and $$\chi _2^l(G)$$χ2l(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, $$\chi _2(G)\le \varDelta +5$$χ2(G)≤Δ+5 if $$4\le \varDelta \le 7$$4≤Δ≤7, and $$\chi _2(G)\le \lfloor \frac{3\varDelta }{2}\rfloor +1$$χ2(G)≤⌊3Δ2⌋+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 $$\chi _2^l(G)\le \varDelta +3$$χ2l(G)≤Δ+3. There exist planar graphs G with girth $$g(G)=6$$g(G)=6 such that $$\chi _2^l(G)=\varDelta +2$$χ2l(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 $$\lambda _l(G)\le \varDelta +8$$λl(G)≤Δ+8 for $$\varDelta \ge 27$$Δ≥27.

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