Edge Coloring with Delays

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.

[1]  N. Alon.  Restricted colorings of graphs , 1993 .

[2]  Michael Molloy,et al.  Near optimal list colorings , 2000 .

[3]  Charles M. Grinstead.  On Medians of Lattice Distributions and a Game with Two Dice , 1997, Combinatorics, Probability and Computing.

[4]  Noga Alon,et al.  A Parallel Algorithmic Version of the Local Lemma , 1991, Random Struct. Algorithms.

[5]  Mark N. Ellingham,et al.  List edge colourings of some 1-factorable multigraphs , 1996, Comb..

[6]  Christian Scheideler,et al.  Coloring non-uniform hypergraphs: a new algorithmic approach to the general Lovász local lemma , 2000, SODA '00.

[7]  B. Bollobás,et al.  Combinatorics, Probability and Computing , 2006 .

[8]  B. Reed.  Graph Colouring and the Probabilistic Method , 2001 .

[9]  Roland Häggkvist,et al.  New Bounds on the List-Chromatic Index of the Complete Graph and Other Simple Graphs , 1997, Combinatorics, Probability and Computing.

[10]  Kenneth Rogers,et al.  A combinatorial problem in Abelian groups , 1963, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  B. Bollobás.  Surveys in Combinatorics , 1979 .

[12]  N. Alon,et al.  The Probabilistic Method, Second Edition , 2000 .

[13]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[14]  Nicholas Bambos,et al.  Scheduling bursts in time-domain wavelength interleaved networks , 2003, IEEE J. Sel. Areas Commun..

[15]  Debasis Mitra,et al.  Light core and intelligent edge for a flexible, thin-layered, and cost-effective optical transport network , 2003, IEEE Commun. Mag..

[16]  Jeff Kahn,et al.  Asymptotically Good List-Colorings , 1996, J. Comb. Theory A.

[17]  Noga Alon.  Combinatorial Nullstellensatz , 1999, Combinatorics, Probability and Computing.

[18]  Artur Czumaj,et al.  Coloring nonuniform hypergraphs: a new algorithmic approach to the general Lovász local lemma , 2000 .

[19]  Penny Haxell,et al.  A Note on Vertex List Colouring , 2001, Combinatorics, Probability and Computing.

[20]  Bruce A. Reed,et al.  Further algorithmic aspects of the local lemma , 1998, STOC '98.

[21]  Noga Alon,et al.  The Probabilistic Method, Second Edition , 2004 .

[22]  Bruce A. Reed,et al.  Asymptotically the List Colouring Constants Are 1 , 2002, J. Comb. Theory, Ser. B.

[23]  József Beck,et al.  An Algorithmic Approach to the Lovász Local Lemma. I , 1991, Random Struct. Algorithms.

[24]  Bruce A. Reed.  The list colouring constants , 1999, J. Graph Theory.