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Suppose k + 1 runners having nonzero constant 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.

Let D be a finite set of integers. The distance graph G(D) has the set of integers as vertices and two vertices at distance d ∈ D are adjacent in G(D). A conjecture of Xuding Zhu states that if the chromatic number of G(D) achieves its maximum value |D| + 1 then the graph has a clique of order |D|. We prove that the chromatic number of a distance graph with D = {a, b, c, d} is five if and only if either D = {1, 2, 3, 4k} or D = {a, b, a + b, a + 2b} with a ≡ 0 (mod 2) and b ≡ 1 (mod 2). This confirms Zhu's conjecture for |D| = 4.

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International audience Let $D$ be a finite set of integers. The distance graph $G(D)$ has the set of integers as vertices and two vertices at distance $d ∈D$ are adjacent in $G(D)$. A conjecture of Xuding Zhu states that if the chromatic number of $G (D)$ achieves its maximum value $|D|+1$ then the graph has a clique of order $|D|$. We prove that the chromatic number of a distance graph with $D=\{ a,b,c,d\}$ is five if and only if either $D=\{1,2,3,4k\}$ or $D=\{ a,b,a+b,a+2b\}$ with $a \equiv 0 (mod 2)$ and $b \equiv 1 (mod 2)$. This confirms Zhu's conjecture for $|D|=4$.