The $$r$$ r -acyclic chromatic number of planar graphs

Abstract

A vertex coloring of a graph G is r-acyclic if it is a proper vertex coloring such that every cycle $$C$$ C receives at least $$\min \{|C|,r\}$$ min { | C | , r } colors. The $$r$$ r -acyclic chromatic number $$a_{r}(G)$$ a r ( G ) of $$G$$ G is the least number of colors in an $$r$$ r -acyclic coloring of $$G$$ G . Let $$G$$ G be a planar graph. By Four Color Theorem, we know that $$a_{1}(G)=a_{2}(G)=\chi (G)\le 4$$ a 1 ( G ) = a 2 ( G ) = ? ( G ) ≤ 4 , where $$\chi (G)$$ ? ( G ) is the chromatic number of $$G$$ G . Borodin proved that $$a_{3}(G)\le 5$$ a 3 ( G ) ≤ 5 . However when $$r\ge 4$$ r ? 4 , the $$r$$ r -acyclic chromatic number of a class of graphs may not be bounded by a constant number. For example, $$a_{4}(K_{2,n})=n+2=\Delta (K_{2,n})+2$$ a 4 ( K 2 , n ) = n + 2 = Δ ( K 2 , n ) + 2 for $$n\ge 2$$ n ? 2 , where $$K_{2,n}$$ K 2 , n is a complete bipartite (planar) graph. In this paper, we give a sufficient condition for $$a_{r}(G)\le r$$ a r ( G ) ≤ r when $$G$$ G is a planar graph. In precise, we show that if $$r\ge 4$$ r ? 4 and $$G$$ G is a planar graph with $$g(G)\ge \frac{10r-4}{3}$$ g ( G ) ? 10 r ? 4 3 , then $$a_{r}(G)\le r$$ a r ( G ) ≤ r . In addition, we discuss the $$4$$ 4 -acyclic colorings of some special planar graphs.