Bounds on the diameter of Cayley graphs of the symmetric group

TL;DR

This paper proves that for certain generating sets of the symmetric group, the maximum word length of permutations is polynomially bounded in n, supporting a conjecture about the diameter of Cayley graphs.

Contribution

It establishes the polynomial bound on the diameter for generating sets containing permutations fixing at least 37% of points, advancing understanding of symmetric group Cayley graphs.

Findings

Polynomial bound on permutation word length proven for specific generator sets

Supports conjecture on Cayley graph diameters of symmetric groups

Provides bounds for sets fixing a significant portion of points

Abstract

In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group of degree n, the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of the points.