# Nonexistence of Efficient Dominating Sets in the Cayley Graphs Generated by Transposition Trees of Diameter 3

## TL;DR

This paper proves that Cayley graphs generated by transposition trees of diameter 3 do not possess efficient dominating sets, extending previous results for graphs with smaller diameters.

## Contribution

It establishes the nonexistence of efficient dominating sets in Cayley graphs generated by transposition trees of diameter 3, a significant extension of prior work for smaller diameters.

## Key findings

- Cayley graphs with diameter less than 3 have efficient dominating sets.
- Cayley graphs generated by diameter 3 trees do not have efficient dominating sets.
- The result generalizes the understanding of dominating sets in these graphs.

## Abstract

Let $d,n$ be positive integers such that $d<n$, and let $X^d_n$ be a Cayley graph generated by a transposition tree of diameter $d$. It is known that every $X^d_n$ with $d<3$ splits into efficient dominating sets. The main result of this paper is that $X^3_n$ does not have efficient dominating sets.