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 colours such that at most $8/15\cdot|E(G)|$ 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.
Highlights
At the ninth British Combinatorial Conference in 1983, Fouquet and Jolivet [4] introduced strong edge-colourings of cubic graphs
The edge e is rich if |f (N (e))| = 4, while it is poor if |f (N (e))| = 2
We demonstrate the upper bound: every bridgeless cubic graph G admits a 4-edgecolouring such that at most
Summary
At the ninth British Combinatorial Conference in 1983, Fouquet and Jolivet [4] introduced strong edge-colourings of cubic graphs. Given an edge-colouring of a cubic graph G, define an edge e to be medium if it is neither rich nor poor. 9], Petersen’s perfect matching theorem combined with Vizing’s edge-colouring theorem (and some further analysis if the graph has cycles of length less than 5) directly yield for every bridgeless cubic graph a 5-edgecolouring such that at least one third of the edges are rich or poor. This bound cannot be improved in general: the Petersen graph has 15 edges and each of its 4-edge-colourings yields at least 8 medium edges This can be checked directly by case analysis (for example by discriminating edge-colourings according to the size of a smallest colour class), but it is rather tedious. We use this formulation of the bound in terms of the number of vertices in the forthcoming proof of Theorem 3
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