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On equitably 2-colourable odd cycle decompositions

Andrea Burgess, Francesca Merola

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 $2\ell$).

On equitably 2-colourable odd cycle decompositions

Abstract

An -cycle decomposition of is said to be \emph{equitably -colourable} if there is a -vertex-colouring of such that each colour is represented (approximately) an equal number of times on each cycle: more precisely, we ask that in each cycle of the decomposition, each colour appears on or of the vertices of . In this paper we study the existence of equitably 2-colourable -cycle decompositions of , where is odd, and prove the existence of such a decomposition for (mod ).
Paper Structure (6 sections, 18 theorems, 38 equations, 1 table)

This paper contains 6 sections, 18 theorems, 38 equations, 1 table.

Table of Contents

  1. Introduction
  2. The $2\ell+1$ and $4\ell+1$ case
  3. Case $v = 2\ell+1$
  4. Case $v=4\ell+1$
  5. Equitably 2-colourable $\ell$-cycle decompositions of $K_v$ when $v \equiv 1$ or $\ell \pmod{2\ell}$
  6. Conclusion

Key Result

Theorem 1.1

There exists a $C_{\ell}$-decomposition of $K_v^*$ if and only if $3 \leq \ell \leq v$ and $\ell$ divides $v\left\lfloor \frac{v-1}{2}\right\rfloor$.

Theorems & Definitions (41)

  • Theorem 1.1: AlspachGavlasSajna
  • Theorem 1.2: BHP
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Example 2.3
  • Theorem 2.4
  • Example 2.5
  • ...and 31 more