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Edge Colouring with Delays

Published online by Cambridge University Press:  01 March 2007

NOGA ALON  and
VERA ASODI
NOGA ALON
Affiliation:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel (e-mail: [email protected])
VERA ASODI
Affiliation:
Department of Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel (e-mail: [email protected])
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Abstract

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(e2) + 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 conjectureholds for some multigraphs, applying algebraic tools.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 2 , March 2007 , pp. 173 - 191
Copyright
Copyright © Cambridge University Press 2006

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