Edge Colouring with Delays

• Authors: Noga Alon;Vera Asodi
• Venue: Combinatorics, Probability and Computing
• Year: 2007

Consider the following communication problem, which leads to a new notion of edge colouring. 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-colouring with delays is a function $f$ from the edges to $\{0,1,\dots,k-1\}$, 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, $f(e_1) + c(e_1) \not \equiv f(e_2) + c(e_2) ~(\mymod~k)$. Ross, Bambos, Kumaran, Saniee and Widjaja [18] conjectured that there always exists a proper edge colouring with delays using $k = \Delta + o(\Delta)$ colours, where $\Delta$ is the maximum degree of the graph. Haxell, Wilfong and Winkler [11] conjectured that a stronger result holds: $k = \Delta + 1$ colours always suffice. We prove the weaker conjecture for simple bipartite graphs, using a probabilistic approach, and further show that the stronger conjectureholds for some multigraphs, applying algebraic tools.