On equitably 2-colourable odd cycle decompositions

TL;DR

This paper investigates the conditions under which complete graphs can be decomposed into odd cycles that are equitably 2-colourable, establishing existence results for certain congruence classes of the number of vertices.

Contribution

It proves the existence of equitably 2-colourable odd cycle decompositions of complete graphs for specific congruence classes of the number of vertices.

Findings

Existence of such decompositions for v ≡ 1, ℓ (mod 2ℓ)

Characterization of equitable 2-colourability in odd cycle decompositions

Extension of previous cycle decomposition results

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 $2\ell$).