Circular flow numbers of regular multigraphs

The circular ow number F c (G) of a graph G = (V; E) is the minimum r 2 Q such that G admits a ow with 1 (e) r ? 1, for each e 2 E. We determine the circular ow number of some regular multigraphs. In particular, we characterize the bipartite (2t + 1)-regular graphs (t 1). Our results imply that there are gaps for possible circular ow numbers for (2t + 1)-regular graphs, e.g. there is no cubic graph G with 3 < F c (G) < 4. We further show that there are snarks with circular ow number arbitrary close to 4, answering a question of X. Zhu.