A vertex coloring is said to be 2-distance if any two distinct vertices of distance at most 2 get different colors. Let $G$ be a planar graph without 4-cycles and 5-cycles. Cranston and Jaeger proved that $G$ is 2-distance $(\Delta(G)+3)$-list colorable if $\Delta(G)\ge 32$. We show that $G$ is 2-distance $(\Delta(G)+2)$-colorable if $\Delta(G)\ge 185760$. The bound $\Delta(G)+2$ is sharp as there exist non-2-distance $(k+1)$-colorable planar graphs of girth 6 and maximum degree $k$ for every integer $k\ge 2$, and there exist non-2-distance $(\Delta(G)+2)$-colorable planar graphs $G$ without 4-cycles or without 5-cycles.

A graph $G$ is said to satisfy the Vizing bound if $\chi(G)\le \omega(G)+1$, where $\chi(G)$ and $\omega(G)$ denote the chromatic number and clique number of $G$, respectively. The class of graphs satisfying the Vizing bound is clearly $\chi$-bounded in the sense of Gyárfás. It has been conjectured that if $G$ is triangle-free and fork-free, where the fork is obtained from $K_{1,4}$ by subdividing two edges, then $G$ satisfies the Vizing bound. We show that this is true if, in addition, $G$ is $C_5$-free.

Given positive integers k and d, a graph G is said to be $(k,d)^*$-colorable if the vertices of G can be colored with k colors such that every vertex has at most d neighbors receiving the same color as itself. Let ${\cal G}$ be the family of plane graphs with neither adjacent triangles nor cycles of length 5. It is proved in this paper that every graph in ${\cal G}$ is $(3,1)^*$-colorable. This result is sharp in the sense that there exist non-$(2,1)^*$-colorable plane graphs with neither triangles nor cycles of length 5. As a corollary, after removing a matching, every graph in ${\cal G}$ is 3-colorable. This provides a partial solution to a conjecture of Borodin and Raspaud [J. Combin. Theory Ser. B, 93 (2003), pp. 17–27].