On equitably 2-colourable odd cycle decompositions

Abstract

An $\ell$-cycle decomposition of $K_v$ is said to be \emph{equitably $2$-colourable} if there is a $2$-vertex-colouring of $K_v$ such that each colour is represented (approximately) an equal number of times on each cycle: more precisely, we ask that in each cycle $C$ of the decomposition, each colour appears on $\lfloor \ell /2\rfloor$ or $\lceil \ell /2\rceil$ of the vertices of $C$. In this paper we study the existence of equitably 2-colourable $\ell$-cycle decompositions of $K_v$, where $\ell$ is odd, and prove the existence of such a decomposition for $v\equiv 1,\ell$ (mod $2\ell$).

24pp