The concept of a line digraph is generalized to that of a directed path graph. The directed path graph → Pk(D) of a digraph D is obtained by representing the directed paths on k vertices of D by vertices. Two vertices are joined by an arc whenever the corresponding directed paths in D form a directed path on k + 1 vertices or form a directed cycle on k vertices in D. Several properties of → P3(...

In this work we consider extensions of a conjecture due to Alspach, Mason, and Pullman from 1976. This concerns edge decompositions tournaments into as few paths possible. There is natural lower bound for the number needed in an decomposition directed graph terms its degree sequence; question states that correct even order. The was recently resolved large tournaments, here investigate what exte...

Certain polymer models are known to exhibit path localization in the sense that at low temperatures, average fractional overlap of two independent samples from Gibbs measure is bounded away $0$. Nevertheless, question where along this takes place has remained unaddressed. In article, we prove on linear scales, occurs entire length polymer. Namely, consider time intervals $\varepsilon N$, $\vare...

A directed path graph is the intersection graph of a family of directed subpaths of a directed tree. A rooted path graph is the intersection graph of a family of directed subpaths of a rooted tree. Rooted path graphs are directed path graphs. Several characterizations are known for directed path graphs: one by forbidden induced subgraphs and one by forbidden asteroids. It is an open problem to ...