On Decomposition of Graphs into Nonisomorphic Independent Sets

A decomposition into non-isomorphic matchings, or \(DINIM\) for short, is a partition of the edges of a graph \(G\) into matchings of different sizes. As a special case of our results, we prove that if a graph \(G\) has at least \((2\chi’ – 2)\chi’ + 1\) edges, where \(\chi’ = \chi'(G)\) is the chromatic index of \(G\), then \(G\) has a \(DINIM\). In particular, the \(n\)-dimensional cube, \(Q_n\), \(n \geq 4\), has a \(DINIM\). These results confirm two conjectures which appeared in Chinn and Richter [3].