On Sequentially Perfect and Sequentially Uniform One Factorizations of Product of Cycles

Let <i>G</i> be an <i>n</i>-regular graph. A sequentially perfect one-factorization is a decomposition of <i>G</i> into one-factors, such that union of any two consecutive one-factors gives a hamiltonian cycle, and also if the union of two consecutive one-factors forms a two regular graph, then this property is termed as sequentially uniform. For any two graphs <i>G</i> and <i>H</i> their product [Cartesian, Tensor, Wreath] has the vertex set {(<i>g, h</i>); <i>g</i> belongs to <i>V</i>(<i>G</i>) and <i>h</i> belongs to <i>V</i>(<i>H</i>)}. The Cartesian Product <i>G</i> Cartesian Product <i>H</i> has the edge set such that two vertices (<i>g<sub>1</sub>, h<sub>1</sub></i>) and (<i>g<sub>2</sub>, h<sub>2</sub></i>) are adjacent if they differ in exactly one coordinate, and the corresponding vertices in that graph are adjacent; in the tensor product <i>G</i> Tensor Product <i>H</i> two vertices (<i>g<sub>1</sub>, h<sub>1</sub></i>) and (<i>g<sub>2</sub>, h<sub>2</sub></i>) are adjacent only if both <i>g<sub>1</sub></i> is adjacent to <i>g<sub>2</sub></i> in <i>G</i> and <i>h<sub>1</sub></i> is adjacent to <i>h<sub>2</sub></i> in <i>H</i>; in the wreath product <i>G</i> Wreath Product <i>H</i> the edge set is <i>E</i>(<i>G</i> Wreath Product <i>H</i>) = {(<i>g<sub>1</sub>, h<sub>1</sub></i>) (<i>g<sub>2</sub>, h<sub>2</sub></i>) : <i>g<sub>1</sub>g<sub>2</sub></i> belongs to <i>E</i>(<i>G</i>) or <i>g<sub>1</sub></i>=<i>g<sub>2</sub></i> and <i>h<sub>1</sub>h<sub>2</sub></i> belongs to <i>E</i>(<i>H</i>)}. In this paper it is proved that for all even <i>m</i> > 2 and odd <i>n</i> ≥ 3, there exists sequentially perfect and sequentially uniform one-factorizations in <i>C<sub>m</sub></i> Cartesian Product <i>C<sub>n</sub></i>, <i>C<sub>m</sub></i> Tensor Product <i>C<sub>n</sub></i> and <i>C<sub>m</sub></i> Wreath Product <i>C<sub>n</sub></i>.