Edge Coloring with Delays

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) ≠ f(e ₂), and for every two edges e ₁ and e ₂ with the same receiver, $f(e_1) + c(e_1) \not \equiv f(e_2) + c(e_2) ~(mod~k)$. Haxell, Wilfong and Winkler [10] conjectured that there always exists a proper edge coloring with delays using k = Δ + 1 colors, where Δ 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.