The binding number of a digraph

Authors:
Genjiu Xu;Xueliang Li;Shenggui Zhang
Affiliations:
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi, P.R. China;Center for Combinatorics and LPMC, Nankai University, Tianjin, P.R. China;Department of Logistics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, P.R. China
Venue:
CJCDGCGT'05 Proceedings of the 7th China-Japan conference on Discrete geometry, combinatorics and graph theory
Year:
2005

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Abstract

Caccetta-Häggkvist's Conjecture discusses the relation between the girth g(D) of a digraph D and the minimum outdegree δċ(D) of D. The special case when g(D) = 3 has lately attracted wide attention. For an undirected graph G, the binding number bind(G) ≥ ??? is a sufficient condition for G to have a triangle (cycle with length 3). In this paper we generalize the concept of binding numbers to digraphs and give some corresponding results. In particular, the value range of binding numbers is given, and the existence of digraphs with a given binding number is confirmed. By using the binding number of a digraph we give a condition that guarantees the existence of a directed triangle in the digraph. The relationship between binding number and connectivity is also discussed.