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.

Source : oai:HAL:hal-00990426v1

Volume: Vol. 12 no. 1

Section: Graph and Algorithms

Published on: January 1, 2010

Submitted on: March 26, 2015

Keywords: acyclic colouring,subcubic graph,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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