Cycles, wheels, and gears in finite planes

TL;DR

This paper proves that all theoretically embeddable cycles can be realized in finite affine and projective planes, and explores embedding wheel and gear graphs in these planes, advancing understanding of their combinatorial structures.

Contribution

It establishes the pancyclicity of finite affine and projective planes and investigates the embedding of wheel and gear graphs within these geometries.

Findings

All cycles embeddable in $AG(2,q)$ and $PG(2,q)$ can be realized.

Finite planes are pancyclic, containing all cycle lengths.

Studied embeddings of wheel and gear graphs in projective planes.

Abstract

The existence of a primitive element of $GF(q)$ with certain properties is used to prove that all cycles that could theoretically be embedded in $AG(2,q)$ and $PG(2,q)$ can, in fact, be embedded there (i.e. these planes are `pancyclic'). We also study embeddings of wheel and gear graphs in arbitrary projective planes.