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

Abstract

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