Graphs with the unique path property: Structure, cycles, factors, and constructions

Abstract

AbstractIn a directed graph with the unique path property, there is a unique directed path of a given length between any ordered pair of its vertices. Such graphs, their paths, cycles, factors, and other properties, were extensively studied during the years, mainly in the context of de Bruijn graphs, forming the most important family of such graphs. Such a graph is equivalent to a binary matrix for which , where is the all‐ones matrix. In this paper, some new results on the structural properties of such graphs, which are not necessarily de Bruijn graphs, are proved. We mainly concentrate on such graphs for which each vertex has in‐degree two and out‐degree two. The minimum number and the maximum number of vertices which are reached from a given vertex using all paths of length smaller than are considered. We provide an exact formula for the number of factors in these graphs and show that this number depends only on the number of alternating cycles in the graph, where in an alternating cycle every two consecutive edges have opposite directions in the cycle. Constructions of such graphs using several methods are presented. In particular, we present a new construction which provides a large set of such graphs with long alternating cycles. We define the integral graph and the derivative graph and implement these definitions on this family of graphs. Various properties from the point of view of the de Bruijn graph and the theory of nonsingular shift‐registers sequences are also considered. Finally, generalizations for graphs in which the in‐degree and the out‐degree of each vertex are greater than two are also discussed.