Maximum Kirkman Signal Sets for Synchronous Uni-Polar Multi-UserCommunication Systems

A unipolar signalingsystem transmits using intensity or amplitude in multiple dimensions.Typical examples arise in optical transmission or radio communicationusing MT-MFSK as both the signaling and the modulation technique.There are dimensions which represent pulses ortones. Each codeword consists of a selection of kof these tones with unit intensity. Each user is assigned mof these binary codewords. In a synchronous multi-user environment,two codewords assigned to a single user have distance 2k,while two codewords assigned to different users have distanceat least 2k-2. Such an assignment of codewords tousers is called a Kirkman signal set when the number of usersaccommodated is the maximum. In this paper, the existence ofKirkman signal sets with k=3 and mas large as possible is settled for all values of .     

On the Hamilton–Waterloo Problem with cycle lengths of distinct parities

Let $K_v^1$ denote the complete graph $K_v$ if $v$ is odd and $K_v?I$, the complete graph with the edges of a 1-factor removed, if $v$ is even. Given non-negative integers $v,M,N,\alpha ,\beta$, the Hamilton–Waterloo problem asks for a 2-factorization of $K_v^1$ into $\alpha$$C_M$-factors and $\beta$$C_N$-factors. Clearly, $M,N\ge 3$, $M\mid v$, $N\mid v$ and $\alpha +\beta =?\frac{v?1}{2}?$ are necessary conditions.Very little is known on the case where $M$ and $N$ have different parities. In this paper, we make some progress on this case by showing, among other things, that the above necessary conditions are sufficient whenever $M|N$, $v>6N>36M$, and $\beta \ge 3$.     

More large sets of resolvable MTS and DTS with even orders

In this paper, we first introduce a special structure that allows us to construct a large set of resolvable Mendelsohn triple systems of orders 2q ＋ 2, or LRMTS（2q ＋ 2）, where q = 6t ＋ 5 is a prime power. Using a computer, we find examples of such structure for t C T = {0, 1, 2, 3, 4, 6, 7, 8, 9, 14, 16, 18, 20, 22, 24}. Furthermore, by a method we introduced in [13], large set of resolvable directed triple systems with the same orders are obtained too. Finally, by the tripling construction and product construction for LRMTS and LRDTS introduced in [2, 20, 21], and by the new results for LR-design in [8], we obtain the existence for LRMTS（v）and LRDTS（v）, where v = 12（t ＋ 1） mi≥0（2.7mi＋1）mi≥0（2.13ni＋1）and t∈T,which provides more infinite family for LRMTS and LRDTS of even orders.     

The Existence of Kirkman Squares—Doubly Resolvable (v,3,1)-BIBDs

A Kirkman square with index , latinicity , block size k, and v points, KS _k (v;,), is a t×t array (t=(v–1)/(k–1)) defined on a v-set V such that (1) every point of V is contained in precisely cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (v, k,)-BIBD. For =1, the existence of a KS _k (v; , ) is equivalent to the existence of a doubly resolvable (v, k, )-BIBD. The spectrum of KS ₂ (v; 1, 1) or Room squares was completed by Mullin and Wallis in 1975. In this paper, we determine the spectrum for a second class of doubly resolvable designs with =1. We show that there exist KS ₃ (v; 1, 1) for , v=3 and v27 with at most 23 possible exceptions for v.     

Steiner triple systems of order 19 and 21 with subsystems of order 7

Steiner triple systems (STSs) with subsystems of order 7 are classified. For order 19, this classification is complete, but for order 21 it is restricted to Wilson-type systems, which contain three subsystems of order 7 on disjoint point sets. The classified STSs of order 21 are tested for resolvability; none of them is doubly resolvable.     

六类Oberwolfach问题OP(4~a,s~b)的解

完全图K_n(n为奇数)或K_n-I(n为偶数,I为K_n的1-因子)是否有2-因子分解称Oberwolfach问题.每个2-因子恰包含α_i个长为m_i的圈(i=1,2,…,t)的Oberwolfach问题记为OP(m_1~(α_1),m_2~(α_2),…,m_t~(α_t)).证明了对任意的a≥0,b=2,3和s=3,5,6,且(a,s,b)≠(0,3,2),都存在OP(4~a,s~b)的解.     

Oberwolfach rectangular table negotiation problem

We completely solve certain case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Let H(k,3) be a bipartite graph with bipartition X={x₁,x₂,…,x_k},Y={y₁,y₂,…,y_k} and edges x₁y₁,x₁y₂,x_ky_k−1,x_ky_k, and x_iy_i−1,x_iy_i,x_iy_i+1 for i=2,3,…,k−1. We completely characterize all complete bipartite graphs K_n,n that can be factorized into factors isomorphic to G=mH(k,3), where k is odd and mH(k,3) is the graph consisting of m disjoint copies of H(k,3).