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\textbf{Multidecompositions of Complete Graphs into a Graph Pair of Order~6}
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Yizhe Gao, Dan Roberts*
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Department of Mathematics, Illinois Wesleyan University, Bloomington, IL 61701
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drobert1@iwu.edu
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A pair of graphs $\{G, H\}$ is a \emph{graph pair of order~$m$} if
(i)~$G$ and $H$ each have order~$m$ and no isolated vertices,
(ii)~$G$ and $H$ are not isomorphic, and
(iii)~$E(G)\cup E(H) = K_m$.
A $(G, H)$-multidecomposition of order~$n$ is a partition of the edges of $K_n$ into copies of $G$ and $H$ with at least one copy of $G$ and at least one copy of~$H$.
We provide necessary and sufficient conditions on $n$ for the existence of a $(G, H)$-multidecomposition of order~$n$ in the case where $G$ is a 6-cycle and $H$ is the complement of a 6-cycle.
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