Circular colorings, orientations, and weighted digraphs

In this paper we prove that if a weighted symmetric digraph (~ G, c) has a mapping T : E( ~ G) → {0, 1} with T (xy) + T (yx) = 1 for all arcs xy in ~ G such that for each dicycle C satisfying 0 < |C|c(mod r) < maxxy∈E(~ G) c(xy) + c(yx) we have |C|c/|C|T ≤ r, then (~ G, c) has a circular r-coloring. Our result generalizes the work of Zhu (J. Comb. Theory, Ser. B, 86 (2002) 109-113) concerning the (k, d)-coloring of a graph, and thus is also a generalization of a corresponding result of Tuza (J. Comb. Theory, Ser. B, 55 (1992) 236-243). Our result also strengthens a result of Goddyn, Tarsi and Zhang (J. Graph Theory 28 (1998) 155-161) concerning the relation between orientation and the (k, d)-coloring of a graph.