On directed and undirected diameters of vertex-transitive graphs
Saveliy V. Skresanov
TL;DR
The paper resolves Babai's conjecture by proving that for every connected vertex-transitive graph on $n$ vertices, the directed diameter satisfies $\overrightarrow{\mathrm{diam}}(\Gamma) = O(\mathrm{diam}(\Gamma)^2 (\log n)^2)$. The approach elevates the problem to the level of homogeneous coherent configurations, proving a general bound for connected relations via a graph-analogue of Ruzsa's triangle inequality and a vertex-expansion framework. A key technical advance is the Ruzsa inequality for graphs, which links the expansion properties of a relation and its symmetrization, enabling a growth argument that forces reachability across the graph. The results also clarify limits by showing that undirected diameter alone cannot bound directed diameter or girth, even for abelian Cayley graphs, via Haight–Ruzsa-type constructions. Overall, the work provides a tight (up to polylog factors) bound for directed diameter in vertex-transitive graphs and introduces graph-specific Ruzsa-type tools of potential independent interest.
Abstract
A directed diameter of a directed graph is the maximum possible distance between a pair of vertices, where paths must respect edge orientations, while undirected diameter is the diameter of the undirected graph obtained by symmetrizing the edges. In 2006 Babai proved that for a connected directed Cayley graph on $n$ vertices the directed diameter is bounded above by a polynomial in undirected diameter and $\log n$. Moreover, Babai conjectured that a similar bound holds for vertex-transitive graphs. We prove this conjecture of Babai, in fact, it follows from a more general bound for connected relations of homogeneous coherent configurations. The main novelty of the proof is a generalization of Ruzsa's triangle inequality from additive combinatorics to the setting of graphs.