Multidesigns for a graph pair of order 6

TL;DR

This paper characterizes the existence and properties of multidecompositions, multipackings, and multicoverings of complete graphs into cycles of length 6 and their complements, providing necessary and sufficient conditions.

Contribution

It provides the first complete characterization of multidecompositions, multipackings, and multicoverings involving $C_6$ and its complement in complete graphs.

Findings

Necessary and sufficient conditions for $(C_6, ar{C}_6)$-multidecomposition of $K_n$.

Characterization of leaves and paddings in maximum multipackings and minimum multicoverings.

Analysis of the cardinalities of leaves and paddings in the specified decompositions.

Abstract

Given two graphs $G$ and $H$, a $(G,H)$-multidecomposition of $K_{n}$ is a partition of the edges of $K_{n}$ into copies of $G$ and $H$ such that at least one copy of each is used. We give necessary and sufficient conditions for the existence of $(C_{6},\overline{C}_{6})$-multidecomposition of $K_{n}$ where $C_{6}$ denotes a cycle of length 6 and $\overline{C}_{6}$ denotes the complement of $C_{6}$. A $(G,H)$-multipacking of $K_{n}$ is a partition of a subset of the edges of $K_{n}$ into copies of $G$ and $H$ such that at least one copy of each is used. The set consisting of the edges of $K_{n}$ that are not used in any copy of either $G$ or $H$ is called the \emph{leave} of the multipacking. A $(G,H)$-multipacking of $K_{n}$ is called \emph{maximum} if the cardinality of the leave is minimum with respect to all $(G,H)$-multipackings of $K_{n}$. A $(G,H)$-multicovering of $K_{n}$ is a $(G,H)$-multidecomposition of $K_{n}$ where some edges can be used repeatedly in copies of $G$ or $H$. The (multi)set of repeated edges is called the \emph{padding} of the $(G,H)$-multicovering of $K_{n}$. A $(G,H)$-multicovering is called \emph{minimum} if the cardinality of the padding is minimum with respect to all $(G,H)$-multicoverings of $K_{n}$. We also characterize the cardinality of the leaves and paddings of maximum $(C_6, \overline{C}_6)$-multipackings and minimum $(C_6, \overline{C}_6)$-multicoverings of $K_{n}$.