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Title:  Edge Coloring with Delays

                 	Vera Asodi
   Center for the Mathematics of Information
       California Institute of Technology

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
k-edge-coloring with delays is a function f from the edges to {0,1,...,k-1},
such that for every two edges e_1 and e_2 with the same transmitter, f(e_1)
\neq f(e_2), and for every two edges e_1 and e_2 with the same receiver,
f(e_1) + c(e_1) \not \equiv f(e_2) + c(e_2) (mod k). Ross et al. conjectured
that there always exists a proper edge coloring with delays using k = \Delta
+ o(\Delta) colors, where \Delta is the maximum degree of the graph. Haxell
et al. conjectured that a stronger result holds:  k = \Delta + 1 colors
always suffice.

In a joint work with Noga Alon, we prove that the stronger conjecture holds
for some multigraphs using algebraic tools, and that the weaker conjecture
holds for all simple bipartite graphs, using a probabilistic approach.

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