Circular Colouring and Orientation of Graphs

This paper proves that if a graph G has an orientation D such that for each cycle C with d∣C∣ (mod k)∈{1,2,…,2d−1} we have ∣C∣/∣C+∣⩽k/d and ∣C∣/∣C−∣⩽k/d, then G has a (k,d)-colouring and hence χc(G)⩽k/d. This is a generalization of a result of Tuza (J. Combin. Theory Ser. B55 (1992), 236–243) concerning the vertex colouring of a graph, and is also a strengthening of a result of Goddyn et al. (J. Graph Theory28 (1998), 155–161) concerning the relation between orientation and circular chromatic number of a graph.