ON DECOMPOSITIONS OF THE COMPLETE EQUIPARTITE GRAPHS K_km(2t)INTO GREGARIOUS m-CYCLES

Abstract

Abstract. For an even integer m at least 4 and any positive integer t, it is shown that the complete equipartite graph K km(2t) can be decomposed into edge-disjoint gregarious m-cycles for any positive integer k under the condition satisfying< k. Here it will be called a gregarious cycle if the cycle has at most one vertex from each partite set.

Gregarious Decompositions of Complete Equipartite Graphs and Related Structures

“…Besides K n * m , complete multipartite graphs with one vertex class of different size were studied. Independently and simultaneously with the preprint version of [12], the 4-cycle systems over K n * m were considered with additional requirements of symmetry in the unpublished manuscript by Cho, Ferrara, Gould, and Schmitt [13] and, later, in the closely related publication by Kim, Cho, and Cho [14].…”

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

“…In this article, we remark that the decompositions in [7] and [11] are circulant, and exhibit some decompositions by examples.…”

“…Given K km (2) , the procedure to produce pairs of f-sequences and flags is explained in [11]. We present two examples in this section following the procedure.…”

“…specified in [11] to both σ ρ and σ λ , we obtain the following four starter cycles. specified in [7] to each of the three s-sequences, we obtain the following six starter cycles.…”

“…The number of edges in K km(2) is 2km(km−1) = 2km(n−1). The author of [11] obtained a γ m -decomposition by producing 2k(n−1) edge-disjoint γ m -cycles in 2k classes, each containing n−1 γ m -cycles.…”