We introduce a sharpened version of the well-known Lonely Runner Conjecture of Wills and Cusick. Given a real number \(x\), let \(\Vert x \Vert\) denote the distance from \(x\) to the nearest integer. For each set of positive integer speeds \(v_1, \dots, v_n\), we define the associated maximum loneliness to be $$\operatorname{ML}(v_1, \dots, v_n)=\max_{t \in \mathbb{R}}\min_{1 \leq i \leq n} \Vert tv_i \Vert.$$

The Lonely Runner Conjecture asserts that \(\operatorname{ML}(v_1, \dots, v_n) \geq 1/(n+1)\) for all choices of \(v_1, \dots, v_n\). We make the stronger conjecture that for each choice of \(v_1, \dots, v_n\), we have either \(\operatorname{ML}(v_1, \dots, v_n)=s/(ns+1)\) for some \(s \in \mathbb{N}\) or \(\operatorname{ML}(v_1, \dots, v_n) \geq 1/n\). This view reflects a surprising underlying rigidity of the Lonely Runner Problem. Our main results are: confirming our stronger conjecture for \(n \leq 3\); and confirming it for \(n=4\) and \(n=6\) in the case where one speed is much faster than the rest.

Mathematics Subject Classifications: 11K60 (primary), 11J13, 11J71, 52C07

Given graphs \(X\) and \(Y\) with vertex sets \(V(X)\) and \(V(Y)\) of the same cardinality, we define a graph \(\mathsf{FS}(X,Y)\) whose vertex set consists of all bijections \(\sigma\colon V(X)\to V(Y)\), where two bijections \(\sigma\) and \(\sigma'\) are adjacent if they agree everywhere except for two adjacent vertices \(a,b \in V(X)\) such that \(\sigma(a)\) and \(\sigma(b)\) are adjacent in \(Y\). This setup, which has a natural interpretation in terms of friends and strangers walking on graphs, provides a common generalization of Cayley graphs of symmetric groups generated by transpositions, the famous \(15\)-puzzle, generalizations of the \(15\)-puzzle as studied by Wilson, and work of Stanley related to flag \(h\)-vectors. We derive several general results about the graphs \(\mathsf{FS}(X,Y)\) before focusing our attention on some specific choices of \(X\). When \(X\) is a path graph, we show that the connected components of \(\mathsf{FS}(X,Y)\) correspond to the acyclic orientations of the complement of \(Y\). When \(X\) is a cycle, we obtain a full description of the connected components of \(\mathsf{FS}(X,Y)\) in terms of toric acyclic orientations of the complement of \(Y\). We then derive various necessary and/or sufficient conditions on the graphs \(X\) and \(Y\) that guarantee the connectedness of \(\mathsf{FS}(X,Y)\). Finally, we raise several promising further questions.

Mathematics Subject Classifications: 05C40, 05C38, 05A05