Embedding Cycles in Finite Planes

Abstract

We define and study embeddings of cycles in finite affine and projective planes. We show that for all $k$, $3\le k\le q^2$,  a $k$-cycle can be embedded in any affine plane of order $q$. We also prove a similar result for finite projective planes: for all $k$, $3\le k\le q^2+q+1$,  a $k$-cycle can be embedded in any projective plane of order $q$.