A remark on cycles through an arc in strongly connected multipartite tournaments

Abstract

A multipartite or c -partite tournament is an orientation of a complete c -partite graph. In this note we prove that a strongly connected c -partite tournament with c ≥ 3 contains an arc that belongs to a directed cycle of length m for every m ∈ { 3 , 4 , … , c } . This result extends those of Bondy [J.A. Bondy, Diconnected orientation and a conjecture of Las Vergnas, J. London Math. Soc. 14 (1976) 277–282] and Yeo [A. Yeo, Diregular c -partite tournaments are vertex-pancyclic when c ≥ 5 , J. Graph Theory 32 (1999) 137–152] for multipartite tournaments and Camion [P. Camion, Chemins et circuits hamiltoniens des graphes complets, C. R. Acad. Sci. Paris 249 (1959) 2151–2152] and Harary and Moser [F. Harary, L. Moser, The theory of round robin tournaments, Amer. Math. Monthly 73 (1966) 231–246] for tournaments.