Abstract
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 χ c ( 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 a graph.
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