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.