TY - JOUR

T1 - Flows, view obstructions, and the lonely runner

AU - Bienia, Wojciech

AU - Goddyn, Luis

AU - Gvozdjak, Pavol

AU - Sebo, András

AU - Tarsi, Michael

N1 - Funding Information:
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 * Research supported by NSERC of Canada. E-mail: goddyn math.sfu.ca. -Research supported by W. Cunningham’s grant from NSERC of Canada. E-mail: sebo imag.fr. Research supported in part by a grant from the Israel Science Foundation. E-mail: tarsi math.tau.ac.il.

PY - 1998

Y1 - 1998

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0002872553&partnerID=8YFLogxK

U2 - 10.1006/jctb.1997.1770

DO - 10.1006/jctb.1997.1770

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AN - SCOPUS:0002872553

SN - 0095-8956

VL - 72

SP - 1

EP - 9

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

IS - 1

ER -