Embedding Factorizations

Abstract

Let $V$ be a set of $n$ vertices for some $n\in\mathbb{N}$ and let $E$ be a collection of $h$-subsets of $V$. Then $\mathscr G = (V,E)$ is an $h$-unifrom hypergraph and we refer to $V$ as its vertex set and to $E$ as its edge set. We say that $\mathscr G$ is complete and denote it by $K_n^h$ if every $h$-subset of $V$ is contained in $E$. If every edge in $E$ is repeated $\lambda$ times, we say $G$ is $\lambda$-fold. Specifically, $\lambda K_n^h$ is the complete $\lambda$-fold $n$-vertex $h$-uniform hypergraph with an edge set containing $\lambda$ copies of every $h$-subset of $V$. In this case, we denote the edge set by $E(\lambda K_n^h)$. Let $\vb r = (r_1, r_2, \dots, r_k)$ for some $r_1, r_2,\dots, r_k \in \mathbb{N}$. An {\it $\vb r$-factorization} of $\lambda K_n^h$ is a partition of $E(\lambda K_n^h)$ into subsets $F_1,\dots, F_k$ such that all elements of $V$ are included at least once in $F_i$ and are included exactly $r_i$ times in $F_i$ for all $i\in\{1,\dots,k\}$. Each such subset $F_i$ is called an $r_i$-factor. A {\it partial $\vb r$-factorization} of $\lambda K_m^h$ is a partition of $E(\lambda K_m^h)$ into $F_1,\dots, F_k$ such that each vertex in $V(\lambda K_m^h)$ is included at most $r_i$ times in each color class $F_i$ for $i \in \{1,\dots,k\}$. Two vertices are adjacent in a hypergraph if some edge in the hypergraph contains both vertices. An $r_i$-factor $F_i$ is connected if for any arbitrary pair of vertices $x,y \in V$, there is some sequence of vertices $x,w_1,w_2,\dots,y$ with each consecutive pair adjacent in $F_i$. In this case, we say that $F_i$ consists of only one component. If we assign some color $i$ to every $h$-subset in $E(\lambda K_n^h)$ for $i\in\{1,\dots,k\}$, we call this a $k$-coloring of $\lambda K_n^h$. An $\vb r$-factorization of $\lambda K_n^h$ is a $k$-coloring of $E(\lambda K_n^h)$ such that edges of each color $i \in \{1,\dots,k\}$ induce an $r_i$-factor. Let $\vb r = (r_1,r_2,\dots,r_q)$ and let $\vb s = (s_1,s_2,\dots,s_k)$ where $r_i, s_j \in \mathbb{N}$ for all $i\in\{1,\dots,q\}, j\in\{1,\dots,k\}$. Motivated by an embedding problem of Peter Cameron and the work of many others, we show that for $n\geq hm$, the obvious necessary conditions that ensure that an $\vb r$-factorization of $\lambda K_m^h$ can be extended to an $\vb s$-factorization of $\lambda K_n^h$ are also sufficient. For $n\geq hm$, we also establish the necessary and sufficient conditions under which an