Path partition in directed graph

Abstract

In this paper, we analyze the bounds of path number in a directed graph, especially in the tournament Tn=(X,U), where we prove that: P(Tn)≤[n24]−2 and from this remarkable result, we give some subclasses of tournaments for which we have P(Tn) = e(Tn) = [n24]−2 (i.e. lower bound of P(G)=upper bound of P(G)) especially if An = (X,U) is the tournament having exactly m−n+1 elementary circuits we have: P(An) = e(An) = [n24]−2. Also we give a general theorem concerning the classes of directed graphs satisfying P(G)=e(G). The result obtained: 1. A procedure that allows a directed graph of order n satisfying P(G)=e(G), another class of directed graph G1 of order n+1 such that P(G1)=e(G1). 2. Generalized certain results proved by Alspach and Ore concerning the path partition number in directed graphs.