View-obstruction: a shorter proof for 6 lonely runners
2004; Elsevier BV; Volume: 287; Issue: 1-3 Linguagem: Inglês
10.1016/j.disc.2004.06.008
ISSN1872-681X
Autores Tópico(s)Limits and Structures in Graph Theory
ResumoIf x is a real number, we denote by 〈x〉∈[0,1) the fractional part of x: 〈x〉=x-E(x), where E(x) is the integer part of x. We give a simple proof of the following version of the Lonely Runner Conjecture: if v1,…,v5 are positive integers, there exists a real number t such that 〈tvi〉∈[16,56] for each i in {1,…,5}. Our proof requires a careful study of the different congruence classes modulo 6 of the speeds v1,…,v5, and is simply based on the consideration of some time t¯ maximizing the distance of 〈tv1〉 to {0,1} among the set of times t such that 〈tvi〉∈[16,56] for each i≠1. In appendix, we also give elementary proofs, based on the same idea, for analogous versions of the conjecture with fewer integers.
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