A connection between circular colorings and periodic schedules

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 [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_uv 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_uv 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 is called a schedule of . Computer scientists are particularly interested in periodic schedules. Given a timed marked graph , they ask if there exist a period p>0 and real numbers x_u such that 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.