Concurrency in heavily loaded neighborhood-constrained systems
ACM Transactions on Programming Languages and Systems (TOPLAS)
Acyclic graph coloring and the complexity of the star chromatic number
Journal of Graph Theory
Circular colorings of weighted graphs
Journal of Graph Theory
Scheduling Parallel Computations
Journal of the ACM (JACM)
Circular chromatic number: a survey
Discrete Mathematics
Homomorphisms to powers of digraphs
Discrete Mathematics - Algebraic and topological methods in graph theory
ANALYSIS OF ASYNCHRONOUS CONCURRENT SYSTEMS BY TIMED PETRI NETS
ANALYSIS OF ASYNCHRONOUS CONCURRENT SYSTEMS BY TIMED PETRI NETS
Concurrency in systems with neighborhood constraints (distributed systems, parallel processing, dining philosophers, simulated annealing)
Resource-sharing system scheduling and circular chromatic number
Theoretical Computer Science
Performance Evaluation of Asynchronous Concurrent Systems Using Petri Nets
IEEE Transactions on Software Engineering
On (k, d)-colorings and fractional nowhere-zero flows
Journal of Graph Theory
Circular colorings of edge-weighted graphs
Journal of Graph Theory
The circular chromatic number of a digraph
Journal of Graph Theory
Journal of Computer and System Sciences
Hi-index | 0.04 |
We show that there is a curious connection between circular colorings of edge-weighted digraphs and periodic schedules of timed marked graphs. Circular coloring of an edge-weighted digraph was introduced by Mohar [B. Mohar, Circular colorings of edge-weighted graphs, J. Graph Theory 43 (2003) 107-116]. This kind of coloring is a very natural generalization of several well-known graph coloring problems including the usual circular coloring [X. Zhu, Circular chromatic number: A survey, Discrete Math. 229 (2001) 371-410] and the circular coloring of vertex-weighted graphs [W. Deuber, X. Zhu, Circular coloring of weighted graphs, J. Graph Theory 23 (1996) 365-376]. Timed marked graphs G- [R.M. Karp, R.E. Miller, Properties of a model for parallel computations: Determinancy, termination, queuing, SIAM J. Appl. Math. 14 (1966) 1390-1411] are used, in computer science, to model the data movement in parallel computations, where a vertex represents a task, an arc uv with weight c"u"v represents a data channel with communication cost, and tokens on arc uv represent the input data of task vertex v. Dynamically, if vertex u operates at time t, then u removes one token from each of its in-arc; if uv is an out-arc of u, then at time t+c"u"v vertex u places one token on arc uv. Computer scientists are interested in designing, for each vertex u, a sequence of time instants {f"u(1),f"u(2),f"u(3),...} such that vertex u starts its kth operation at time f"u(k) and each in-arc of u contains at least one token at that time. The set of functions {f"u:u@?V(G-)} is called a schedule of G-. Computer scientists are particularly interested in periodic schedules. Given a timed marked graph G-, they ask if there exist a period p0 and real numbers x"u such that G- has a periodic schedule of the form f"u(k)=x"u+p(k-1) for each vertex u and any positive integer k. In this note we demonstrate an unexpected connection between circular colorings and periodic schedules. The aim of this note is to provide a possibility of translating problems and methods from one area of graph coloring to another area of computer science.