ON THE ERD\H{O}S-GYÁRFÁS CONJECTURE FOR SOME CAYLEY GRAPHS

Abstract

In 1995, Paul Erd\H{o}s and András Gyárfás conjectured thatfor every graph $X$ of minimum degree at least $3$, there exists a non-negative integer $m$ such that $X$ contains a simple cycle of length $2^m$.In this paper, we prove that the conjecture holds for Cayley graphs of order $2p^2$ and $4p$.