Abstract
We prove that the edges of every even graph G=G 1+G 2 that is the join of two regular graphs G 1 and G 2 can be coloured with Δ(G) colours, whenever Δ(G)=Δ(G 1)+|V 2|. The proof of this result together with the results in De Simone and Galluccio (J. Comb. Optim. 18:417–428, 2009) states that every even graph G that is the join of two regular graphs is Class 1.
The proof yields an efficient combinatorial algorithm to find a Δ(G)-edge-colouring of this type of graphs.
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De Simone, C., Galluccio, A. Edge-colouring of joins of regular graphs II. J Comb Optim 25, 78–90 (2013). https://doi.org/10.1007/s10878-011-9420-4
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DOI: https://doi.org/10.1007/s10878-011-9420-4