On a Class of Highly Symmetric k-Factorizations

Abstract

A $k$-factorization of $K_v$ of type $(r, s)$ consists of $k$-factors each of which is the disjoint union of $r$ copies of $K_{k+1}$ and $s$ copies of $K_{k,k}$. By means of what we call the patterned $k$-factorization $F_k(D)$ over an arbitrary group $D$ of order $2s + 1$, it is shown that a $k$-factorization of type $(1, s)$ exists for any $k\ge2$ and for any $s\ge1$ with $D$ being an automorphism group acting sharply transitively on the factor-set. The general method to construct a $k$-factorization $F$ of type $(1, s)$ over an arbitrary 1-factorization $S$ of $K_{2s+2}$ ($F$ is said to be based on $S$) is used to prove that the number of pairwise non-isomorphic $k$-factorizations of this type goes to infinity with $s$. In this paper, we show that the full automorphism group of $F$ is known as soon as we know the one of $S$. In particular, the full automorphism group of $F_k(D)$ is determined for any $k\ge2$, generalizing a result given by P. J. Cameron for patterned 1-factorizations [J London Math Soc 11 (1975), 189-201]. Finally, it is shown that $F_k(D)$ has exactly $(k!)2s+1(2s+1)|Aut(D)|$ automorphisms whenever $D$ is abelian.