On oriented path double covers

Abstract

In this paper we concentrate on an oriented version of perfect path double cover (PPDC). An oriented perfect path double cover (OPPDC) of a graph G is a collection of oriented paths in the symmetric orientation GS of G such that each edge of GS lies in exactly one of the paths and for each vertex v of G there is a unique path which begins in v (and thus the same holds also for terminal vertices of the paths). First we show that the graphs K3 and K5 have no OPPDC. Then we study the structure of a minimal connected graph G ≠ K3, G ≠ K5 which has no OPPDC either. We show that the minimal degree in this graph is at least 4. © 2001 Elsevier Science B.V. All rights reserved.