Constructing an infinite family of cubic 1-regular graphs (Q1864571)

An arc in a graph is an edge fixed with one of the two directions. A graph is 1-regular if there is an automorphism that carries any arc to any other arc. In this paper, the authors construct an infinite family of cubic 1-regular graphs of order \(8(k^2 + k + 1)\), where \(k \geq 2\). It was known that there were cubic 1-regular graphs of order \(2n\), when \(n \geq 13\) is odd and 3 divides \(\phi (n)\) (\(\phi\) denotes the Euler phi function). The authors show that their new class is different from this known class. Their graphs are constructed using cyclic-covering graphs of the 3-dimensional cube.