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arXiv · 1411.0856

Graphs with few paths of prescribed length between any two vertices

David Conlon
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Abstract

We use a variant of Bukh's random algebraic method to show that for every natural number $k \geq 2$ there exists a natural number $\ell$ such that, for every $n$, there is a graph with $n$ vertices and $Ω_k(n^{1 + 1/k})$ edges with at most $\ell$ paths of length $k$ between any two vertices. A result of Faudree and Simonovits shows that the bound on the number of edges is tight up to the implied constant.

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David Conlon. 2019-08-16. Graphs with few paths of prescribed length between any two vertices. https://doi.org/10.1112/blms.12295

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