 
Summary: Edge Coloring with Delays
Noga Alon
Vera Asodi
Abstract
Consider the following communication problem, that leads to a new notion of edge coloring.
The communication network is represented by a bipartite multigraph, where the nodes on one
side are the transmitters and the nodes on the other side are the receivers. The edges correspond
to messages, and every edge e is associated with an integer c(e), corresponding to the time it
takes the message to reach its destination. A proper kedgecoloring with delays is a function
f from the edges to {0, 1, ..., k  1}, such that for every two edges e1 and e2 with the same
transmitter, f(e1) = f(e2), and for every two edges e1 and e2 with the same receiver, f(e1) +
c(e1) f(e2) + c(e2) (mod k). Haxell, Wilfong and Winkler [10] conjectured that there always
exists a proper edge coloring with delays using k = + 1 colors, where is the maximum degree
of the graph. We prove that the conjecture asymptotically holds for simple bipartite graphs, using
a probabilistic approach, and further show that it holds for some multigraphs, applying algebraic
tools. The probabilistic proof provides an efficient algorithm for the corresponding algorithmic
problem, whereas the algebraic method does not.
1 Introduction
Motivated by the study of optical networks, Haxell, Wilfong and Winkler considered in [10] a commu
nication network in which there are two groups of nodes: transmitters and receivers. Each transmitter
