2 research outputs found
A note on the simultaneous edge coloring
Let be a graph. A (proper) -edge-coloring is a coloring of the
edges of such that any pair of edges sharing an endpoint receive distinct
colors. A classical result of Vizing ensures that any simple graph admits a
-edge coloring where denotes the maximum degreee of
. Recently, Cabello raised the following question: given two graphs
of maximum degree on the same set of vertices , is it
possible to edge-color their (edge) union with colors in such a way
the restriction of to respectively the edges of and the edges of
are edge-colorings? More generally, given 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
of maximum degree with 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 colors are enough which is, as far as we know, the
best known upper bound