# Acyclic edge-colouring of planar graphs

1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : A proper edge-colouring with the property that every cycle contains edges of at least three distinct colours is called an {\it acyclic edge-colouring}. The {\it acyclic chromatic index} of a graph $G$, denoted $\chi'_a(G)$ is the minimum $k$ such that $G$ admits an {\it acyclic edge-colouring} with $k$ colours. We conjecture that if $G$ is planar and $\Delta(G)$ is large enough then $\chi'_a(G)=\Delta(G)$. We settle this conjecture for planar graphs with girth at least $5$ and outerplanar graphs. We also show that $\chi'_a(G)\leq \Delta(G) + 25$ for all planar graph $G$, which improves a previous result by Muthu et al.
Document type :
Reports

https://hal.inria.fr/inria-00367394
Contributor : Frederic Havet <>
Submitted on : Wednesday, March 11, 2009 - 10:47:16 AM
Last modification on : Monday, October 12, 2020 - 10:30:20 AM
Long-term archiving on: : Friday, October 12, 2012 - 1:25:24 PM

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RR-6876.pdf
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• HAL Id : inria-00367394, version 1

### Citation

Nathann Cohen, Frédéric Havet, Tobias Mueller. Acyclic edge-colouring of planar graphs. [Research Report] RR-6876, INRIA. 2009. ⟨inria-00367394⟩

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