Dedicated to Mieczysław Borowiecki on the occasion of his 70th birthday

## Acyclic 6-colouring of graphs with maximum degree 5 and small maximum average degree

 Anna Fiedorowicz Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra Z. Szafrana 4a, 65-516 Zielona Góra, Poland

## Abstract

A k-colouring of a graph G is a mapping c from the set of vertices of G to the set {1, …,k} of colours such that adjacent vertices receive distinct colours. Such a k-colouring is called acyclic, if for every two distinct colours i and j, the subgraph induced by all the edges linking a vertex coloured with i and a vertex coloured with j is acyclic. In other words, every cycle in G has at least three distinct colours.

Acyclic colourings were introduced by Grünbaum in 1973, and since then have been widely studied. In particular, the problem of acyclic colourings of graphs with bounded maximum degree has been investigated. In 2011, Kostochka and Stocker showed that any graph with maximum degree 5 can be acyclically coloured with at most 7 colours. The question, whether this bound is achieved, remains open. In this note we prove that any graph with maximum degree 5 and maximum average degree at most 4 admits an acyclic 6-colouring. We also provide examples of graphs with these properties.

Keywords: acyclic colouring, bounded degree graph, maximum average degree

2010 Mathematics Subject Classification: 05C15.

## References

 [1] O.V. Borodin, On acyclic colorings of planar graphs, Discrete Math. 25 (1979) 211--236, doi: 10.1016/0012-365X(79)90077-3 . [2] O.V. Borodin, A.V. Kostochka and D.R. Woodall, Acyclic colorings of planar graphs with large girth, J. Lond. Math. Soc. 60 (1999) 344--352, doi: 10.1112/S0024610799007942. [3] M.I. Burstein, Every 4-valent graph has an acyclic 5-coloring, Soobšč. Akad. Gruzin. SSR 93 (1979) 21--24 (in Russian). [4] G. Fertin and A. Raspaud, Acyclic coloring of graphs of maximum degree five: Nine colors are enough, Inform. Process. Lett. 105 (2008) 65--72, doi: 10.1016/j.ipl.2007.08.022. [5] B. Grünbaum, Acyclic coloring of planar graphs, Israel J. Math. 14 (1973) 390--408, doi: 10.1007/BF02764716. [6] A.V. Kostochka, Upper bounds of chromatic functions of graphs, Ph.D. Thesis, Novosibirsk, 1978 (in Russian). [7] A.V. Kostochka and C. Stocker, Graphs with maximum degree 5 are acyclically 7-colorable, Ars Math. Contemp. 4 (2011) 153--164. [8] S. Skulrattanakulchai, Acyclic colorings of subcubic graphs, Inform. Process. Lett. 92 (2004) 161--167, doi: 10.1016/j.ipl.2004.08.002. [9] D. West, Introduction to Graph Theory, 2nd ed. (Prentice Hall, Upper Saddle River, 2001).