Toward a Three-dimensional Counterpart of Cruse's Theorem

Completing partial latin squares is NP-complete. Motivated by Ryser's theorem for latin rectangles, in 1974, Cruse found conditions that ensure a partial symmetric latin square of order $m$ can be embedded in a symmetric latin square of order $n$. Loosely speaking, this results asserts that an $n$-coloring of the edges of the complete $m$-vertex graph $K_m$ can be embedded in a one-factorization of $K_n$ if and only if $n$ is even and the number of edges of each color is at least $m-n/2$. We establish necessary and sufficient conditions under which an edge-coloring of the complete $\lambda$-fold $m$-vertex 3-graph $\lambda K_m^3$ can be embedded in a one-factorization of $\lambda K_n^3$. In particular, we prove the first known Ryser type theorem for hypergraphs by showing that if $n \equiv 0 \;(\bmod\; 3)$, any edge-coloring of $\lambda K_m^3$ where the number of triples of each color is at least $m/2-n/6$, can be embedded in a one-factorization of $\lambda K_n^3$. Finally we prove an Evans type result by showing that if $n \equiv 0 \;(\bmod\; 3)$ and $n\geq 3m$, then any $q$-coloring of the edges of any $F\subseteq\lambda K_m^3$ can be embedded in a one-factorization of $\lambda K_n^3$ as long as $q\leq \lambda \binom{n-1}{2}-\lambda \binom{m}{3}/\left\lfloor m/3 \right\rfloor$.