Cycles of arbitrary length in distance graphs on $\mathbb{F}_q^d$

TL;DR

This paper investigates the existence of cycles of arbitrary length in distance and dot-product graphs over finite fields, establishing conditions on the size of the vertex set for their guaranteed presence.

Contribution

It proves that large enough subsets of finite vector spaces contain the expected number of cycles in distance graphs, and extends results to dot-product graphs with more advanced methods.

Findings

Distance graphs contain many cycles if |E| ≥ C_k q^{(d+2)/2}.

Results extend to dot-product graphs with more complex techniques.

Longer cycles require smaller size thresholds for guaranteed existence.

Abstract

For $E \subset {\Bbb F}_q^d$, $d \ge 2$, where ${\Bbb F}_q$ is the finite field with $q$ elements, we consider the distance graph ${\mathcal G}^{dist}_t(E)$, $t \not=0$, where the vertices are the elements of $E$, and two vertices $x$, $y$ are connected by an edge if $||x-y|| \equiv {(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2=t$. We prove that if $|E| \ge C_k q^{\frac{d+2}{2}}$, then ${\mathcal G}^{dist}_t(E)$ contains a statistically correct number of cycles of length $k$. We are also going to consider the dot-product graph ${\mathcal G}^{prod}_t(E)$, $t \not=0$, where the vertices are the elements of $E$, and two vertices $x$, $y$ are connected by an edge if $x \cdot y \equiv x_1y_1+\dots+x_dy_d=t$. We obtain similar results in this case using more sophisticated methods necessitated by the fact that the function $x \cdot y$ is not translation invariant. The exponent $\frac{d+2}{2}$ is improved for sufficiently long cycles.