Chen and Chvatal's conjecture in tournaments

Abstract

In this work we present a version of the so called Chen and Chv´atal’s conjecture for directed graphs. A line of a directed graph D is defined by an ordered pair (u, v), with u and v two distinct vertices of D, as the set of all vertices w such that u, v, w belong to a shortest directed path in D containing a shortest directed path from u to v. A line is empty if there is no directed path from u to v. Another option is that a line is the set of all vertices. The version of the Chen and Chv´atal’s conjecture we study states that if none of previous options hold, then the number of distinct lines in D is at least its number of vertices. Our main result is that any tournament satisfies this conjecture as well as any orientation of a complete bipartite graph of diameter three.