A note on the simultaneous edge coloring

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 (1964) ensures that any simple graph G admits a (Delta(G) + 1)-edge coloring where Delta(G) denotes the maximum degree of G. Recently, Cabello raised the following question: given two graphs G(1), G(2) of maximum degree Delta on the same set of vertices V, is it possible to edge-color their (edge) union with Delta + 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 Delta with Omega(root l center dot Delta) 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 3/2 Delta + 4 colors are enough which is, as far as we know, the best known upper bound. (C) 2019 Elsevier B.V. All rights reserved.