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Acyclic colourings of graphs with bounded degree

Abstract : A k-colouring of a graph G is called acyclic if for every two distinct colours i and j, the subgraph induced in G by all the edges linking a vertex coloured with i and a vertex coloured with j is acyclic. In other words, there are no bichromatic alternating cycles. In 1999 Boiron et al. conjectured that a graph G with maximum degree at most 3 has an acyclic 2-colouring such that the set of vertices in each colour induces a subgraph with maximum degree at most 2. In this paper we prove this conjecture and show that such a colouring of a cubic graph can be determined in polynomial time. We also prove that it is an NP-complete problem to decide if a graph with maximum degree 4 has the above mentioned colouring.
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Mieczyslaw Borowiecki, Anna Fiedorowicz, Katarzyna Jesse-Jozefczyk, Elzbieta Sidorowicz. Acyclic colourings of graphs with bounded degree. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2010, 12 (1), pp.59-74. ⟨hal-00990426⟩

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