×

zbMATH — the first resource for mathematics

Out-arc pancyclicity of vertices in tournaments. (English) Zbl 1220.05049
Summary: T. Yao, Y. Guo and K. Zhang [“Pancyclic out-arcs of a vertex in a tournament,” Discrete Appl. Math. 99, No. 1-3, 245–249 (2000; Zbl 0939.05045)] proved that every strong tournament contains a vertex \(u\) such that every out-arc of \(u\) is pancyclic. In this paper, we prove that every strong tournament with minimum out-degree at least two contains two such vertices. A. Yeo [“The number of pancyclic arcs in a \(k\)-strong tournament,” J. Graph Theory 50, No. 3, 212–219 (2005; Zbl 1081.05041)] conjectured that every 2-strong tournament has three distinct vertices \(\{ x,y,z \}\), such that every arc out of \(x,y\) and \(z\) is pancyclic. In this paper, we also prove that Yeo’s conjecture is true.

MSC:
05C20 Directed graphs (digraphs), tournaments
05C38 Paths and cycles
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bang-Jensen, J.; Gutin, G., Digraphs: theory, algorithms and applications, (2000), Springer London
[2] Camion, P., Chemins et circuits hamiltoniens des graphes complets, C.R. acad. sci. Paris, 249, 2151-2152, (1959) · Zbl 0092.15801
[3] Feng, J., Each 3-strong tournament contains 3 vertices whose out-arcs are pancyclic, Graphs combin., 25, 299-307, (2009) · Zbl 1207.05070
[4] Li, R.; Li, S.; Feng, J., The number of vertices whose out-arcs are pancyclic in a 2-strong tournament, Discrete appl. math., 156, 88-92, (2008) · Zbl 1126.05053
[5] Thomassen, C., Hamiltonian-connected tournaments, J. combin. theory ser. B, 28, 142-163, (1980) · Zbl 0435.05026
[6] Yao, T.; Guo, Y.; Zhang, K., Pancyclic out-arcs of a vertex in a tournament, Discrete appl. math., 99, 245-249, (2000) · Zbl 0939.05045
[7] Yeo, A., The number of pancyclic arcs in a \(k\)-strong tournament, J. graph theory, 50, 212-219, (2005) · Zbl 1081.05041
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.