Long paths in the distance graph over large subsets of vector spaces over finite fields
Abstract
Let $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements. Construct a graph, called the distance graph of $E$, by letting the vertices be the elements of $E$ and connect a pair of vertices corresponding to vectors $x,y \in E$ by an edge if $||x-y||={(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2=1$. We shall prove that if the size of $E$ is sufficiently large, then the distance graph of $E$ contains long non-overlapping paths and vertices of high degree.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2014
- DOI:
- 10.48550/arXiv.1406.0107
- arXiv:
- arXiv:1406.0107
- Bibcode:
- 2014arXiv1406.0107B
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Number Theory;
- 52C10