Graph Decompositions of \(K(v, \lambda)\) into Modified Triangles Using Langford and Skolem Sequences

Abstract

We decompose the complete multigraph \(K(v, \lambda)\) into copies of a graph \(H_i\) (\(i = 1, 2, 3\)). Each \(H_i\) is a near-triangle in that it is connected and has \(3\) vertices. In several cases, the decompositions are completed using classical combinatorial sequences due to Langford and Skolem.