Balanced Quatrefoil Decomposition of Complete Multigraphs

Definition. Let Kn denote the complete graph of n vertices. The complete multigraph λKn is the complete graph Kn in which every edge is taken λ times. The t-foil (or the t-windmill) is a graph of t edge-disjoint K3’s with a common vertex and the common vertex is called the center of the t-foil. In particular, the 2-foil, the 3-foil, and the 4-foil are called the bowtie, the trefoil, and the quatrefoil, respectively. When λKn is decomposed into edge-disjoint sum of t-foils, we say that λKn has a t-foil decomposition. Moreover, when every vertex of λKn appears in the same number of t-foils, we say that λKn has a balanced t-foil decomposition and this number is called the replication number. This balanced t-foil decomposition of λKn is called a balanced t-foil design. (Fig. 1) Ushio and Fujimoto [12] gave the necessary and sufficient condition for the existence of the balanced bowtie decomposition of λKn is n ≥ 5 and λ(n − 1) ≡ 0 (mod 12). Decomposition algorithms were also given. The purpose of this paper is to give the necessary and sufficient condition for the existence of the balanced quatrefoil decomposition of λKn together with its decomposition algorithms. In this paper, it is shown that the necessary and sufficient condition for the existence of a balanced quatrefoil decomposition of λKn is n ≥ 9 and λ(n − 1) ≡ 0 (mod 24). As one of problems that are well known in the field of mathematical programming, there is the Vehicle Routing