A note on the simultaneous edge coloring
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Let G=(V,E) be a graph. A (proper) k-edge-coloring is a coloring of the edges of G such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph G admits a (Δ(G)+1)-edge coloring where Δ(G) denotes the maximum degreee of G. Recently, Cabello raised the following question: given two graphs G_1,G_2 of maximum degree Δ on the same set of vertices V, is it possible to edge-color their (edge) union with Δ+2 colors in such a way the restriction of G to respectively the edges of G_1 and the edges of G_2 are edge-colorings? More generally, given ℓ graphs, how many colors do we need to color their union in such a way the restriction of the coloring to each graph is proper? In this short note, we prove that we can always color the union of the graphs G_1,...,G_ℓ of maximum degree Δ with Ω(√(ℓ)·Δ) colors and that there exist graphs for which this bound is tight up to a constant multiplicative factor. Moreover, for two graphs, we prove that at most 3/2Δ +4 colors are enough which is, as far as we know, the best known upper bound.
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