Resolvable Decompositions of \(\lambda K_n\) into the Union of Two \(2\)-Paths

Abstract

We give necessary and sufficient conditions for a resolvable \(4\)-decomposition of \(AK_n\), in the case where \(H\) is one of the 10 graphs obtained by the union of two paths of length 2, with two possible exceptions. In particular, we complete the \(4\)-star (\(\lambda\)) and \(T\) (\(\tau\)) for higher \(\lambda\) and give complete solutions for resolvable decompositions into Fish (\(4\)-\(3\)), Mulinetto (\(hx\)) and Kites (\(BSI\)). In the cases of the Fish and Mulinetto the solution is obtained \(1\)-rotationally.