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

TL;DR

This paper investigates the structure of the distance graph over large subsets of finite field vector spaces, demonstrating the existence of long paths and high-degree vertices when the subset size is sufficiently large.

Contribution

It establishes new results on the existence of long paths and vertices of high degree in the distance graph over large finite field subsets.

Findings

Existence of long non-overlapping paths in the distance graph

Presence of vertices with high degree in the graph

Results depend on the size of the subset being sufficiently large

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.