Concurrency in heavily loaded neighborhood-constrained systems
ACM Transactions on Programming Languages and Systems (TOPLAS)
On the complexity of H-coloring
Journal of Combinatorial Theory Series B
A note on the star chromatic number
Journal of Graph Theory
Some aspects of the parallel and distributed iterative algorithms—a survey
Automatica (Journal of IFAC)
Star chromatic numbers and products of graphs
Journal of Graph Theory
Acyclic graph coloring and the complexity of the star chromatic number
Journal of Graph Theory
Journal of Combinatorial Theory Series B
The star chromatic number of a graph
Journal of Graph Theory
Circular colorings of weighted graphs
Journal of Graph Theory
Multichromatic numbers, star chromatic numbers and Kneser graphs
Journal of Graph Theory
Circular chromatic number: a survey
Discrete Mathematics
Circular colouring and orientation of graphs
Journal of Combinatorial Theory Series B
On the fractional chromatic number, the chromatic number, and graph products
Discrete Mathematics
Optimal Mapping of Neighbourhood-Constrained Systems
IRREGULAR '95 Proceedings of the Second International Workshop on Parallel Algorithms for Irregularly Structured Problems
Circular chromatic number of Kneser graphs
Journal of Combinatorial Theory Series B
Concurrency in systems with neighborhood constraints (distributed systems, parallel processing, dining philosophers, simulated annealing)
On (k, d)-colorings and fractional nowhere-zero flows
Journal of Graph Theory
The circular chromatic number of series-parallel graphs
Journal of Graph Theory
Randomized generation of acyclic orientations upon anonymous distributed systems
Journal of Parallel and Distributed Computing
Note: A connection between circular colorings and periodic schedules
Discrete Applied Mathematics
Interleaved adjoints of directed graphs
European Journal of Combinatorics
Resource-sharing systems and hypergraph colorings
Journal of Combinatorial Optimization
Scheduling links for heavy traffic on interfering routes in wireless mesh networks
Computer Networks: The International Journal of Computer and Telecommunications Networking
Hi-index | 5.23 |
A graph G is used as a model for a resource sharing system, where each vertex represents a process and an edge joining two vertices means that the corresponding processes share a resource. A scheduling of G is a mapping f : {1, 2, 3, ... } → 2V(G), where f(i) consists of processes that are operating at round i. The rate of f is defined as rate(f) = lim supn → ∞ Σi=1n|f(i)|/n|V(G)|, which is the average fraction of operating processes at each round. A scheduling is fair if adjacent vertices alternate their turns in operating. The operating rate γ*(G) of G is the maximum rate of a fair scheduling. Fair schedulings of a graph was first studied by Barbosa and Gafni. They introduced the method of "scheduling by edge reversal" which derives a fair scheduling through an acyclic orientation. Through scheduling by edge reversal, Barbosa and Gafni related γ* (G) to the structure of acyclic orientations of G. We point out that this relation implies that γ* (G) is equal to the reciprocal of the circular chromatic number of G. Both circular coloring and scheduling by edge reversal have been studied extensively in the past decade. The former by graph theorists, and the latter by computer scientists. However, it seems that neither side knew the existence of the other. This paper intends to build a connection between the two sides. We show that certain open problems concerning scheduling by edge reversal are indeed solved under the language of circular coloring. In the study of fair scheduling, Barbosa and Gafni defined a variation of multiple coloring of graphs: the interleaved p-color, q-tuple colorings. We formulate the interleaved coloring as a graph homomorphism problem. In the study of circular chromatic number, Bondy and Hell defined (p, q)-colorings and also formulated it as a graph homomorphism problem. We prove that the target graph for the interleaved p-color, q-tuple coloring and the target graph of (p, q)-coloring are homomorphically equivalent. This gives another proof of the fact that γ* (G) = 1/χc (G). Moreover, the proof gives an explicit formula which deduces an optimal circular coloring of G from an optimal interleaved coloring of G, and vice versa. This paper also introduces two other schedulings of a graph, the weakly fair scheduling and the strongly fair scheduling. It is proved that the rate of an optimal strongly fair scheduling of a graph G is also equal to the reciprocal of the circular chromatic number of G, and the rate of an optimal weakly fair scheduling of G is equal to the reciprocal of the fractional chromatic number of G. Barbosa and Gafni presented an algorithm that determines the rate γo(w) of the scheduling induced by an acyclic orientation ω of G. By using Karp's minimum mean cycle algorithm, we give a faster algorithm to calculate γo(w).