Geodesics in Oriented Graphs

Abstract

A geodesic from a to b in a directed graph is the shortest directed path from a to b . R. C. Entringer made the beautiful conjecture that every graph has an orientation in which every geodesic is unique. Unfortunately this is not true; the aim of this note is to present two different ways of disproving it. First, we show that for a large enough q the Paley graph Q q , has no orientation in which every geodesic is unique, and then we apply a probabilistic argument to show that almost no graph has an orientation with unique geodesies. I am grateful to Adrian Bondy for bringing Entringer's conjecture to my attention. Let q be a prime power and write F q , for the field of order q . If q is a power of a prime congruent to 1 modulo 4, then the Paley graph Q q , is defined on F q , by joining a to b if a – b is a quadratic residue. It is well known that Q q is a strongly regular graph of degree ( q – 1)/2, in which two adjacent vertices have ( q – 1)/4 common neighbours. For the sake of completeness we give detailed proofs of the following folklore lemmas concerning Paley graphs. I am grateful to J. W. S. Cassels for the slick proof of the first.