Discussiones Mathematicae Graph Theory 33(1) (2013) 91-99
doi: 10.7151/dmgt.1665

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

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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


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.


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Received 14 July 2012
Revised 14 November 2012
Accepted 15 November 2012