Infinitely many nonsolvable groups whose Cayley graphs are hamiltonian

Dave Witte Morris

doi:10.13069/jacodesmath.66457

Abstract

We show there are infinitely many finite groups~$G$, such that every connected Cayley graph on~$G$ has a hamiltonian cycle, and $G$ is not solvable. Specifically, we show that if $A_5$~is the alternating group on five letters, and $p$~is any prime, such that $p \equiv 1 \pmod{30}$, then every connected Cayley graph on the direct product $A_5 \times \integer _p$ has a hamiltonian cycle.

Highlights

Throughout this paper, all rings are associative with identity
An element of a ring R is called nil-clean if it is the sum of an idempotent and a nilpotent element
Breaz et al in [1] proved their main result that the matrix ring Mn(F ) over a field F is nil-clean if and only if F ∼= F2, where F2 is the field of two elements

Summary

Introduction

Throughout this paper, all rings are associative with identity. An element in a ring R is said to be (strongly) clean if it is the sum of an idempotent and a unit element(and these commute ). Abstract: An element of a ring R is called nil-clean if it is the sum of an idempotent and a nilpotent element. We show that the n × n matrix ring over a principal ideal domain R is a nil-clean ring if and only if R is isomorphic to F2. We show that the same result is true for the 2 × 2 matrix ring over an integral domain R.

Results

Conclusion