Circular colouring and orientation of graphs

Abstract

This paper proves that if a graph G has an orientation D such that for each cycle C with d C (mod k) ε {1,2,⋯,2d - 1} we have C / C+ ≤k/d and C / C- ≤k/d, then G has a (k,d)-colouring and hence Xc(G)≤k/d. This is a generalization of a result of Tuza (J. Combin. Theory Ser. B 55 (1992), 236-243) concerning the vertex colouring of a graph, and is also a strengthening of a result of Goddyn et al. (J. Graph Theory 28 (1998), 155-161) concerning the relation between orientation and circular chromatic number of graph. © 2002 Elsevier Science (USA).