A note on the simultaneous edge coloring

Abstract

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 (Δ(G)+1)-edge coloring where Δ(G) denotes the maximum degree of G. Recently, Cabello raised the following question: given two graphs G1,G2 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 G1 and the edges of G2 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 G1,…,Gℓ 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 32Δ+4 colors are enough which is, as far as we know, the best known upper bound.