双重准可分解4-圈系

Doubly Near Resolvable 4-Cycle Systems

设G=λKv是λ重v阶完全图,即任意一对顶点间恰有λ条边相连.图G的一个m-圈系是长度为m的圈的集合C,其中所有圈的边恰好构成图G边集的划分.若C中的m-圈能够划分成为准平行类R={R1,R2,…,Rλv/2},那么就称该m-圈系C为v阶准可分解m-圈系,记为(v,m,λ)-NRCS,且称R为该设计的一个准分解类.如果(v,m,λ)-NRCS C存在一对正交准分解类,则称之为双重准可分解m-圈系,记为(v,m,λ)-DNRCS.当m=2和3时,(v,2,2)-DNRCS以1v型Room方和(v,3,2)-DNRBIBD为大家所知.Mullin和Wallis建立了1v型Room方存在的谱系.Abel、Lamken、Vanstone和Wang等建立了(v,3,2)-DNRBIBD存在的谱系.文章利用直接构作和递推构作完全建立了(v,4,2)-DNRCS存在的谱系.即证明了(v,4,2)-DNRCS存在的充分必要条件是v≡1(mod 4),其中v=9是唯一例外.

Let G = λKvbe the complete graph on v vertices in which each pair of vertices is joined by exactly λedges. An m-cycle system of G is a collection C of cycles of length m whose edges partition the edges of G. An mcycle system C of λKnis said to be near resolvable if the m-cycles in C can be partitioned into near parallel classes R = {R1, R2, …, Rλv/2} and C is denoted by（v, m, λ）-NRCS, R is called a near resolution. If a（v, m, λ）-NRCS has a pair of orthogonal near resolutions, it is named as doubly resolvable and is denoted by（v, m, λ）-DNRCS. For m = 2, 3,（v, m, 2）-DNRCSs are known as Room squares of type 1vand（v, 3, 2）-DNRBIBDs, respectively.Mullin and Wallis had established the spectrum for Room squares of type 1v. Abel_Lamken_Vanstone and Wang had established the spectrum for（ v, 3, 2）- DNRBIBDs. In this paper, applying direct constructions and recursive constructions, the spectrum of the existence of（v, 4, 2）-DNRCSs is established, i.e. It is shown that there exists a（v, 4, 2）-DNRCS, if and only if v≡1（mod 4） with definite exception v = 9.