Hamilton cycles for involutions of classical types

Abstract

Let ${\mathcal W}_n$ denote any of the three families of classical Weyl groups: the symmetric groups ${\mathcal S}_n$, the hyperoctahedral groups (signed permutation groups) ${\mathcal S}^B_n$, or the even-signed permutation groups ${\mathcal S}^D_n$. In this paper we give an uniform construction of a Hamilton cycle for the restriction to involutions on these three families of groups with respect to a inverse-closed connecting set of involutions. This Hamilton cycle is optimal with respect to the Hamming distance only for the symmetric group ${\mathcal S}_n$. We also recall an optimal algorithm for a Gray code for type $B$ involutions. A modification of this algorithm would provide a Gray Code for type $D$ involutions with Hamming distance two, which would be optimal. We give such a construction for ${\mathcal S}^D_4$ and ${\mathcal S}^D_5$.

Figures (2)

• Figure 1: Dynkin diagrams for irreducible finite Weyl groups.
• Figure 5: An Hamilton path in $G(I_3^D)$ with Hamming distance 2.

Theorems & Definitions (20)

• Proposition 2.1
• proof
• Example 3.1
• Example 3.2
• Lemma 3.1
• Theorem 3.2
• proof
• Corollary 3.3
• proof
• Theorem 3.4