Pancyclic out-arcs of a Vertex in Tournaments

Thomassen (J. Combin. Theory Ser. B 28, 1980, 142–163) proved that every strong tournament contains a vertex x such that each arc going out from x is contained in a Hamiltonian cycle. In this paper, we extend the result of Thomassen and prove that a strong tournament contains a vertex x such that every arc going out from x is pancyclic, and our proof yields a polynomial algorithm to nd such a vertex. Furthermore, as another consequence of our main theorem, we get a result of Alspach (Canad. Math. Bull. 10, 1967, 283–286) that states that every arc of a regular tournament is pancyclic. ? 2000 Elsevier Science B.V. All rights reserved. 1. Terminology and introduction We denote the vertex set and the arc set of a digraph D by V (D) and E(D), respectively. A subdigraph induced by a subset A⊆V (D) is denoted by D[A]. In addition, D − A= D[V (D)− A]. If xy is an arc of a digraph D, then we say that x dominates y and write x → y. We also say that xy is an out-arc of x or xy is an in-arc of y. More generally, if A and B are two disjoint subdigraphs of D such that every vertex of A dominates every vertex of B, then we say that A dominates B and write A→ B. Let x be a vertex of D. The number of out-arcs of x is called the out-degree of x and denoted by dD(x), or simply d +(x). Note that a tournament Tn is regular if and only if all vertices of Tn have the same out-degree. We consider only the directed paths and cycles. A digraph D is strong if for every pair of vertices x and y; D contains a path from x to y and a path from y to x, and ( This work was supported by NSFC under no. 19471037. ∗ Corresponding author. fax: +49 241 8888 390. E-mail address: [email protected] (Y. Guo) 1 The author is kindly supported by a grant from “Deutsche Forschungsgemeinschaft” as a member of “Graduiertenkolleg: Analyse und Konstruktion in der Mathematik” at RWTH Aachen. 0166-218X/00/$ see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S0166 -218X(99)00136 -5 246 T. Yao et al. / Discrete Applied Mathematics 99 (2000) 245–249 D is k-connected if |V (D)|¿k+1 and for any set A⊂V (D) of at most k−1 vertices, D − A is strong. A k-cycle is a cycle of length k. An arc of a digraph on n¿3 vertices is said to be pancyclic if it is contained in a k-cycle for all k satisfying 36k6n. In [4], Thomassen proved that every strong tournament contains a vertex x such that each out-arc of x is contained in a Hamiltonian cycle and this extends the result of Goldberg and Moon [2] that every s-strong tournament has at least s distinct Hamiltonian cycles (a digraph D is s-strong if for any set F ⊂E(D) of at most s− 1 arcs, D − F is strong). In this paper we extend the result of Thomassen and prove that every strong tournament contains a vertex x such that all out-arcs of x are pancyclic, and our proof yields a polynomial algorithm to nd such a vertex. In addition, as another consequence of our main theorem, we get a result of Alspach [1] that states that every arc of a regular tournament is pancyclic.