Variations on the Petersen colouring conjecture

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

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