In this article, we prove the existence of a simple cyclic near-resolvable v-1/2-cycle system of 2Kv for v≡9  mod  12 by the method of constructing its starter. Then, some new properties and results related to this construction are formulated.

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A compatible factorization of order , is an array in which the entries in row form a near-one-factor with focus , and the triples associated with the rows contain no repetitions. In this paper, we aim to amend this compatible factorization so that we can display triples with the minimum repeated triples. Throughout this paper we propose a new type of factorization called near-compatible factorization. First, we present the compatible factorization towards developing a near-compatible factorization. Second, we discuss briefly the necessary and sufficient conditions for the existence of near-compatible factorization. Then, we exemplify the construction for case as a groundwork in developing near-compatible factorization.

In combinatorial design theory, clustering elements into a set of three elements is the heart of classifying data. This article will provide insight into formulating algorithm for a new type of triple system, called a Butterfly triple system. Basically, in this algorithm development, a starter of cyclic near-resolvable ((v-1)/2)-cycle system of the 2-fold complete graph 2K_v is employed to construct the starter of cyclic ((v-1)/2)-star decomposition of 2K_v. These starters were then decomposed into triples and classified as a starter of a cyclic Butterfly triple system. The obtained starter set generated a triple system of order A special reference for case 𝑣𝑣 ≡ 9 (mod 12) was presented to demonstrate the development of the Butterfly triple system.