A Remark on Circulant Decompositions of Complete Multipartite Graphs by Gregarious Cycles

Abstract

Abstract. Let k, m and t be positive integers with m ≥ 4 and even. It is shown that K km+1(2t) has a decomposition into gregarious m-cycles. Also, it is shown that K km(2t) has a decomposition into gregarious m-cycles if< k. In this article, we make a remark that the decompositions can be circulant in the sense that it is preserved by the cyclic permutation of the partite sets, and we will exhibit it by examples.

Gregarious Decompositions of Complete Equipartite Graphs and Related Structures

“…The case where the cycle length equals the number of classes turns out to be easier to handle and admits resolvable decompositions, as proven by Billington, Hoffman, and Rodger [19]. General cycle lengths have also been considered by Smith [20], Kim [21], and Cho [22,23]. A common feature of those results is that, for any fixed k, they provide an infinite family of constructions for every fixed k, allowing arbitrarily large class size m (and also, an arbitrarily large number n of classes, except in [19]).…”