Graphs with few paths of prescribed length between any two vertices
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 $\Omega_k(n^{1 + 1/k})$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2014
- DOI:
- 10.48550/arXiv.1411.0856
- arXiv:
- arXiv:1411.0856
- Bibcode:
- 2014arXiv1411.0856C
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 8 pages