Non‐path spectrum sets

Abstract

A path of a graph is called maximal if it is not a proper subpath of any other path of the graph. The path spectrum of a graph G, denoted by ps (G), is the set of lengths of all maximal paths in the graph. A set S of positive integers is called a path spectrum if there is a connected graph G such that ps(G) = S. Jacobson et al. showed that all sets of positive integers with cardinality of 1 or 2 are path spectrum sets. Their results raised the question of whether all sets of positive integers are path spectra. We show that, for every positive integer k ≥ 3, there are infinitely many sets of k positive integers which are not path spectra. A set S of positive integers is called an absolute path spectrum if there are infinitely many connected