A few more Lonely Runners

2022/11/16 by Avinash Bhardwaj, Bhardwaj, Avinash, Vishnu Narayanan +3
Computer Science · Mathematics · #51M20 #52C07 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2211.08749

openalex publication_date 2022/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Lonely Runner Conjecture, proposed by Jörg M. Wills and so nomenclatured by Luis Goddyn, has been an object of interest since it was first conceived in 1967 : Given positive integers k and n₁,n₂,…,n_k there exists a positive real number t such that the distance of t⋅ n_j to the nearest integer is at least (1)/(k+1), ∀~~1≤ j≤ k. In a recent article Beck, Hosten and Schymura described the Lonely Runner polyhedron and provided a polyhedral approach to identifying families of lonely runner instances. We revisit the Lonely Runner polyhedron and highlight some new families of instances satisfying the conjecture. In addition, we relax the sufficiency of existence of an integer point in the Lonely Runner polyhedron to prove the conjecture. Specifically, we propose that it suffices to show the existence of a lattice point of certain superlattices of the integer lattice in the Lonely Runner polyhedron.

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