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arXiv · 2312.12943

On directed and undirected diameters of vertex-transitive graphs

Saveliy V. Skresanov
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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.

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Saveliy V. Skresanov. 2023-12-20. On directed and undirected diameters of vertex-transitive graphs. https://doi.org/10.1007/s00493-024-00120-4

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