Journal of Combinatorial Optimization

, Volume 29, Issue 4, pp 713–722

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

• Guanghui Wang
• Guiying Yan
• Jiguo Yu
• Xin Zhang
Article

## Abstract

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

## Keywords

Acyclic coloring Planar graph Girth

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## Authors and Affiliations

• Guanghui Wang
• 1
Email author
• Guiying Yan
• 2
• Jiguo Yu
• 3
• Xin Zhang
• 4
1. 1.School of MathematicsShandong UniversityJinanPeople’s Republic of China
2. 2.Academy of Mathematics and System SciencesChinese Academy of SciencesBeijingPeople’s Republic of China
3. 3.School of Computer ScienceQufu Normal UniversityRizhaoPeople’s Republic of China
4. 4.Department of MathematicsXidian UniversityXianPeople’s Republic of China