TY - JOUR

T1 - Edge colouring with delays

AU - Alon, Noga

AU - Asodi, Vera

N1 - Funding Information:
†Research supported in part by a USA–Israeli BSF grant, by the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. ‡A preliminary version of this paper appeared in RANDOM-APPROX 2004, Vol. 3122 of Lecture Notes in Computer Science, Springer, pp. 237–248.
Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2007/3

Y1 - 2007/3

N2 - 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,..., 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(e 2) + c(e2) (mod k). Ross, Bambos, Kumaran, Saniee and Widjaja [18] conjectured that there always exists a proper edge colouring with delays using k = Δ + o(Δ) colours, where Δ is the maximum degree of the graph. Haxell, Wilfong and Winkler [11] conjectured that a stronger result holds: k = Δ + 1 colours always suffice. We prove the weaker conjecture for simple bipartite graphs, using a probabilistic approach, and further show that the stronger conjecture holds for some multigraphs, applying algebraic tools.

AB - 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,..., 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(e 2) + c(e2) (mod k). Ross, Bambos, Kumaran, Saniee and Widjaja [18] conjectured that there always exists a proper edge colouring with delays using k = Δ + o(Δ) colours, where Δ is the maximum degree of the graph. Haxell, Wilfong and Winkler [11] conjectured that a stronger result holds: k = Δ + 1 colours always suffice. We prove the weaker conjecture for simple bipartite graphs, using a probabilistic approach, and further show that the stronger conjecture holds for some multigraphs, applying algebraic tools.

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U2 - 10.1017/S0963548306007851

DO - 10.1017/S0963548306007851

M3 - Article

AN - SCOPUS:33847172698

VL - 16

SP - 173

EP - 191

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 2

ER -