Flows, View Obstructions, and the Lonely Runner

We prove the following result. Let G be an undirected graph. If G has a nowhere zero ow with at most k di erent values, then it also has one with values from the set f1; : : : ; kg. When k 5, this is a trivial consequence of Seymour's \sixow 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 (Montash. Math. 71 (1967)) and independently by T. Cusick (Aequationes Math. 9 (1973)). Fortunately for our purposes, this conjecture has been veri ed 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 self-contained proof of the case k = 4 of the Lonely Runner Conjecture. AMS Classi cations (1991): 11J13, (05B35, 05C15, 05C50, 11J71, 11K60, 52C07, 90B10)