On equitably 2-colourable odd cycle decompositions

An l $\ell $-cycle decomposition of K v ${K}_{v}$ is said to be equitably 2-colourable if there is a 2-vertex-colouring of K v ${K}_{v}$ such that each colour is represented (approximately) an equal number of times on each cycle: more precisely, we ask that in each cycle C $C$ of the decomposition, each colour appears on L l / 2 RIGHT FLOOR $\lfloor \ell \unicode{x02215}2\rfloor $ or left ceiling l / 2 right ceiling $\lceil \ell \unicode{x02215}2\rceil $ of the vertices of C $C$. In this paper we study the existence of equitably 2-colourable l $\ell $-cycle decompositions of K v ${K}_{v}$, where l $\ell $ is odd, and prove the existence of such a decomposition for v equivalent to 1 , l $v\equiv 1,\ell $ (mod 2 l $2\ell $).