On arc-traceable local tournaments

• Published in: Discrete mathematics Vol. 308; no. 24; pp. 6513 - 6526
• Main Authors: Meierling, Dirk, Volkmann, Lutz
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 28.12.2008; Elsevier
• Subjects: Applied sciences; Arc-traceable; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Exact sciences and technology; Graph theory; Hamiltonian path; Information retrieval. Graph; Local tournaments; Mathematics; Sciences and techniques of general use; Theoretical computing; Local tournaments; Hamiltonian path; Arc-traceable; Vertex; Sufficient condition; Digraph; Tournament; Directed graph
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2007.12.042

A digraph D is arc-traceable if for every arc x y of D , the arc x y belongs to a directed Hamiltonian path of D . A local tournament is an oriented graph such that the negative neighborhood as well as the positive neighborhood of every vertex induces a tournament. It is well known that every tournament contains a directed Hamiltonian path and, in 1990, Bang-Jensen showed the same for connected local tournaments. In 2006, Busch, Jacobson and Reid studied the structure of tournaments that are not arc-traceable and consequently gave various sufficient conditions for tournaments to be arc-traceable. Inspired by the article of Busch, Jacobson and Reid, we develop in this paper the structure necessary for a local tournament to be not arc-traceable. Using this structure, we give sufficient conditions for a local tournament to be arc-traceable and we present examples showing that these conditions are best possible.