Optimal Edge Coloring of Large Graphs

k-lnterval labeling schemes (k-ILS) are compact routing schemes on general networks which have been studied extensively and recently been implemented on the latest generation INMOS Transputer Router chips. In this paper, we introduce an extension of the k-ILS to the (k, s)-DFILS (deadlock-free ILS), where k is the number of intervals and s is the number of buffers used at each node or edge to prevent deadlock. Whereas k-ILS only compactly represents shortest paths between pairs of nodes, this new extension aims to represent those particular ones that give rise also to deadlock-free routing controllers which use a low number of buffers per node or per edge. In particular, we consider deadlock-free routing controllers obtained using a standard deadlock prevention technique (acyclic orientation coverings) which can be applied both to packet and wormhole routing. While both time and space complexity results are given for general networks, tight results are shown for specific topologies, such as trees, rings, grids, complete graphs, and chordal rings. Moreover, trade-offs are derived between the number of intervals k and the number of buffers s in (k, s)-DFILS for hypercubes, grids, tori, and Cartesian products of graphs. © 1999 John Wiley & Sons, Inc. Networks 34: 47-60, 1999