TY - CHAP

T1 - Edge coloring with delays

AU - Alon, Noga

AU - Asodi, Vera

PY - 2004

Y1 - 2004

N2 - 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 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). 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.

AB - 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 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). 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.

UR - http://www.scopus.com/inward/record.url?scp=35048888981&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35048888981&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-27821-4_22

DO - 10.1007/978-3-540-27821-4_22

M3 - Chapter

AN - SCOPUS:35048888981

SN - 3540228942

SN - 9783540228943

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 237

EP - 248

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

A2 - Jansen, Klaus

A2 - Khanna, Sanjeev

A2 - Rolim, Jose D. P.

A2 - Ron, Dana

PB - Springer Verlag

ER -