Variations on the Petersen colouring conjecture

Variations sur la conjecture de coloration par le graphe de Petersen

François Pirot (1, 2) , Jean-Sébastien Sereni (3) , Riste Škrekovski (4)

1 ORPAILLEUR - Knowledge representation, reasonning
2 Radboud University [Nijmegen]
3 CNRS - Centre National de la Recherche Scientifique
4 University of Ljubljana - Department of Mathematics

The Petersen colouring conjecture states that every bridgeless cubic graph admits an edge-colouring with 5 colours such that for every edge e, the set of colours assigned to the edges adjacent to e has cardinality either 2 or 4 but not 3. We prove that every bridgeless cubic graph G admits an edge-colouring with 4 such that at most 4 |V(G)| / 5 edges do not satisfy the above condition. This bound is tight and the Petersen graph is the only connected graph for which the bound cannot be decreased. We obtain such a 4-edge-colouring by using a carefully chosen subset of edges of a perfect matching, and the analysis relies on a simple discharging procedure with essentially no reductions and very few rules.

Domaines: Combinatoire [math.CO]

François Pirot, Jean-Sébastien Sereni, Riste Škrekovski. Variations on the Petersen colouring conjecture. 2018. ⟨hal-02133840v1⟩