This paper proves that if a graph G has an orientation D such that for each cycle C with d jCj ðmod kÞ 2 f1; 2; . . . ; 2d 1g we have jCj=jCþj4k=d and jCj=jC j4k=d; then G has a ðk; dÞ-colouring and hence wcðGÞ4k=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… (More)