Flows, view obstructions, and the lonely runner

Abstract

We prove the following result: Let G be an undirected graph. If G has a nowhere zero flow with at most k different values, then it also has one with values from the set {1, ..., k}. When k ≥ 5, this is a trivial consequence of Seymour's "six-flow theorem". When k ≤ 4 our proof is based on a lovely number theoretic problem which we call the "Lonely Runner Conjecture:" Suppose k runners having nonzero constant speeds run laps on a unit-length circular track. Then there is a time at which all runners are at least 1/(k+1) from their common starting point. This conjecture appears to have been formulated by J. Wills (Monatsch. Math. 71, 1967) and independently by T. Cusick (Aequationes Math. 9, 1973). This conjecture has been verified for k≤4 by Cusick and Pomerance (J. Number Theory 19, 1984) in a complicated argument involving exponential sums and electronic case checking. A major part of this paper is an elementary selfcontained proof of the case k = 4 of the Lonely Runner Conjecture. © 1998 Academic Press.