Cycles, wheels, and gears in finite planes
  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. 
Jamie Peabody, Oscar Vega and Jordan White
math.CO