Cycle decompositions of cartesian products of two cycles

Abstract

We say a graph $H$ decomposes a graph $G$ if there exists a partition of the edges of $G$ into subgraphs isomorphic to $H$. We seek to characterize necessary and sufficient conditions for a cycle of length $k$, denoted $C_k$, to decompose the Cartesian product of two cycles $C_m ~\square~ C_n$. We prove that if $m$ is a multiple of 3, then the Cartesian product of a cycle $C_m$ and any other cycle can be decomposed into 3 cycles of equal length. This extends work of Kotzig, who proved in 1973 that the Cartesian product of two cycles can always be decomposed into two cycles of equal length. We also show that if $k$, $m$, and $n$ are positive, and $k$ divides $4mn$ then $C_{4k}$ decomposes $C_{4m} ~\square~ C_{4n}$.