Hamilton Cycles in Restricted and Incomplete Rotator Graphs

Combinatorial Gray codes-an updated survey

“…Compton and Williamson [CW93], as an application of their doubly-adjacent Gray code, showed that we can also use the generators {σ n , σ −1 n , σ 2 }. Stevens and Williams [SW12b] showed the same result for the generators {σ n , σ 3 , σ 2 }, and Ruskey and Williams [RW10] proved it for {σ n , σ n−1 }; see Figure 8 (k). In a recent breakthrough, Sawada and Williams [SW20] give a simple and explicit rule to generate all permutations of length n using only the two generators {σ n , σ 2 }, and this listing is cyclic if and only if n is odd; see Figure 1 (e) and Figure 8 (l).…”

“…P32 Despite all of the aforementioned results, in general it is wide open for which sets of generators X the Cayley graph Γ(S n , X) or the Cayley digraph #" Γ (S n , X) have a Hamilton cycle or path. Generalizing the sigma-tau problem, we may consider X = {σ n , (1 k)} or X = {σ n , σ k } for suitable values of n and 2 ≤ k ≤ n. Specifically, the case X = {σ n , σ 3 } for even n was suggested by Stevens and Williams [SW12b]. Furthermore, Savage [Sav97] mentions X = {ρ n , ρ n−1 , (n − 1 n)} as an interesting open problem.…”

Combinatorial Gray Codes — an Updated Survey

“…P31 Despite all of the aforementioned results, in general it is wide open for which sets of generators X the Cayley graph Γ(S n , X) or the Cayley digraph #" Γ (S n , X) have a Hamilton cycle or path. Generalizing the sigma-tau problem, we may consider X = {σ n , (1 k)} or X = {σ n , σ k } for suitable values of n and 2 ⩽ k ⩽ n. Specifically, the case X = {σ n , σ 3 } for even n was suggested by Stevens and Williams [SW12b]. Furthermore, Savage [Sav97] mentions X = {ρ n , ρ n−1 , (n − 1 n)} as an interesting open problem.…”

Hamiltonicity of the Cayley Digraph on the Symmetric Group Generated by σ = (1 2 ... n) and τ = (1 2)

“…Prior to this article, Hamilton cycles of the undirected cayley(S n , {σ, τ }) were constructed with great difficultly by Compton and Williamson [1]. Hamilton cycles in − −−−− → cayley(S n , G) have also been constructed for G = {σ, τ, (1 2 3)} by Stevens and Williams [10], and G = {σ, (1 2 ••• n−1)} by Holroyd, Ruskey, and Williams [3]. The literature frequently states that G(5) is not Hamiltonian; see Ruskey, Jiang, and Weston [8] for the history and resolution of this error.…”