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Variations on the Petersen colouring conjecture [Elektronski vir]
Pirot, François ; Sereni, Jean-Sébastien ; Škrekovski, Riste
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 colours such that at most 8|E(G)|/15 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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