On Hamilton cycles in connected tetravalent metacirculant graphs with non-empty first symbol (Q1431395)

Metacirculant graphs were introduced by \textit{B.~Alspach} and \textit{T.~D.~Parsons} [Can. J. Math. 34, 307-318 (1982; Zbl 0467.05032)] as an interesting class of vertex-transitive graphs. The purpose of the present paper is to prove that every connected \((m,n)\)-metacirculant of degree four with non-empty first symbol contains a Hamilton cycle, provided \(m\in\{1,2\}\) or \(m>2\) and both \(m\) and \(n\) are odd. This extends previous work of the first author [J. Graph Theory 23, 273-287 (1996; Zbl 0869.05037)] on the Hamiltonicity of connected metacirculants of degree four which are not Cayley graphs.