The Lonely Runner with Seven Runners

2008/03/20 by Javier Barajas, Oriol Serra · 44 citations
Computer Science · Mathematics · #Advanced Graph Theory Research #Graph Labeling and Dimension Problems #Computational Geometry and Mesh Generation #Conjecture #Combinatorics #Mathematics #Constant (computer programming) #Context (archaeology) #Chromatic scale #Pairwise comparison #Lonely runner conjecture #Computer science #Statistics #Geography #Collatz conjecture

paper · pdf · doi:10.37236/772

published in The Electronic Journal of Combinatorics 15(1) (Electronic Journal of Combinatorics)

openalex publication_date 2008/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/27

Abstract

Suppose k+1 runners having nonzero constant pairwise distinct speeds run laps on a unit-length circular track starting at the same time and place. A runner is said to be lonely if she is at distance at least 1/(k+1) along the track to every other runner. The lonely runner conjecture states that every runner gets lonely. The conjecture has been proved up to six runners (k≤ 5). A formulation of the problem is related to the regular chromatic number of distance graphs. We use a new tool developed in this context to solve the first open case of the conjecture with seven runners.

Citations

• Flows, View Obstructions, and the Lonely Runner
• View-obstruction problems, III
• View-obstruction problems
• View-obstruction problems in n-dimensional geometry
• View-obstruction: a shorter proof for 6 lonely runners
• On the chromatic number of circulant graphs
• Six Lonely Runners

Cited by

• The lonely runner problem for many runners
• Barely lonely runners and very lonely runners: a refined approach to the Lonely Runner Problem
• Correlation Among Runners and Some Results on the Lonely Runner Conjecture
• Lonely runner conjecture [wikipedia]