Generalized fibonacci maximum path graphs

Abstract

We investigate the following problem: Given integers m and n, find an acyclic directed graph with m edges and n vertices and two distinguished vertices s and t such that the number of distinct paths from s to t (not necessarily disjoint) is maximized. It is shown that there exists such a graph containing a Hamiltonian path, and its structure is investigated.We give a complete solution to the cases (i) m⩽2n−3 and (ii) m = kn−12k(k+1)+r for k =1, 2, …, n− and r=0,1,2.