zbMATH — the first resource for mathematics

Circulant tournaments of prime order are tight. (English) Zbl 1198.05083
Summary: We say that a tournament is tight if for every proper 3-coloring of its vertex set there is a directed cyclic triangle whose vertices have different colors. In this paper, we prove that all circulant tournaments with a prime number \(p\geq 3\) of vertices are tight using results relating to the acyclic disconnection of a digraph and theorems of additive number theory.

05C20 Directed graphs (digraphs), tournaments
05C05 Trees
Full Text: DOI
[1] Alspach, B., Cycles of each length in regular tournaments, Canad. math. bull., 10, 283-285, (1967) · Zbl 0148.43602
[2] Arocha, J.; Bracho, J.; Neumann-Lara, V., On the minimum size of tight hypergraphs, J. graph theory, 16, 4, 319-326, (1992) · Zbl 0776.05079
[3] Arocha, J.; Bracho, J.; Neumann-Lara, V., Tight and untight triangulated surfaces, J. combin. theory (B), 63, 185-199, (1995) · Zbl 0832.05035
[4] Bondy, J.A.; Murty, U.S.R., Graph theory with applications, (1976), Elsevier New York · Zbl 1134.05001
[5] Galeana-Sánchez, H.; Neumann-Lara, V., A class of tight circulant tournaments, Discuss. math. graph theory, 20, 109-128, (2000) · Zbl 0969.05031
[6] Goddard, W.D.; Kubicki, G.; Oellermann, O.R.; Tian, S., On multipartite tournaments, J. combin. theory ser. B, 52, 2, 284-300, (1991) · Zbl 0736.05040
[7] Nathanson, M.B., ()
[8] Neumann-Lara, V., The acyclic disconnection of a digraph, Discrete math., 197-198, 617-632, (1999) · Zbl 0928.05033
[9] Vosper, A.G., The critical pairs of subsets of a group of prime order, J. London math. soc., 31, 200-205, (1956) · Zbl 0072.03402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.