Circulant homogeneous factorisations of complete digraphs \(\mathbf K_{p^{d}}\) with \(p\) an odd prime

Jing Xu

The electronic journal of combinatorics, Tome 24 (2017) no. 2

Résumé

Let $\mathcal F=(\rm\bf K_{n},\mathcal P)$ be a circulant homogeneous factorisation of index $k$, that means $\mathcal P$ is a partition of the arc set of the complete digraph $\rm\bf K_n$ into $k$ circulant factor digraphs such that there exists $\sigma\in S_n$ permuting the factor circulants transitively amongst themselves. Suppose further such an element $\sigma$ normalises the cyclic regular automorphism group of these circulant factor digraphs, we say $\mathcal F$ is normal. Let $\mathcal F=(\rm\bf K_{p^d},\mathcal P)$ be a circulant homogeneous factorisation of index $k$ where $p^d$, ($d\ge 1$) is an odd prime power. It is shown in this paper that either $\mathcal F$ is normal or $\mathcal F$ is a lexicographic product of two smaller circulant homogeneous factorisations.

Détail

DOI : 10.37236/6477

Classification : 05C25, 20B25
Mots-clés : circulant homogeneous factorisations, normal circulant homogeneous factorisations, lexicographic product