A note on hamiltonicity conditions of the coprime and non-coprime graphs of a finite group

Let G be a group. The coprime and non-coprime graphs of G are introduced by Ma et al. (2014) and Mansoori et al. (2016), respectively, when G is finite. By their definitions, which refer to coprime and non-coprime terms of two positive integers, those graphs must be related. We prove that they are closely related through their graph complement and preserve the isomorphism groups. Furthermore, according to Cayley's theorem, which states that any group G is isomorphic to a subgroup of the symmetric group on G, it implies that the studies of the coprime and non-coprime graphs of any group G (especially, when G is finite) can actually be represented by the coprime and non-coprime graphs of any subgroup of the symmetric group on G. This encourages us to specifically study the hamiltonicity of both kinds of graphs associated with G when G is isomorphic to the symmetric group on G.