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arXiv · 1406.0107

Long paths in the distance graph over large subsets of vector spaces over finite fields

M. Bennett·J. Chapman·D. Covert·D. Hart·A. Iosevich·J. Pakianathan
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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.

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M. Bennett, J. Chapman, D. Covert, D. Hart, A. Iosevich, J. Pakianathan. 2014-05-31. Long paths in the distance graph over large subsets of vector spaces over finite fields. https://arxiv.org/abs/1406.0107

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