Qingde Kang

Large sets of oriented triple systems with resolvability

Abstract A triple system (X, B ) is called to be resolvable (or almost resolvable), if B can be partitioned into parallel classes (or almost parallel classes), where the parallel class (or almost parallel class) is a partition of X (or X ⧹{ x } for some x ∈ X ). In this paper, we summarize the present situation about large sets (resp. overlarge sets) of several types of triple systems and give some new results.

Large sets of Hamilton cycle and path decompositions

The existence spectrums for large sets of Hamilton cycle decompositions and Hamilton path decompositions are completed. Also, we show that the completion of large sets of directed Hamilton cycle decompositions and directed Hamilton path decompositions depends on the existence of certain special tuscan squares. Several conjectures about special tuscan squares are posed.

On overlarge sets of Kirkman triple systems

The determination of the existence of large sets of Kirkman triple systems (LKTS) is a classical combinatorial problem. In this paper we introduce a new concept OLGKS-overlarge sets of generalized Kirkman triple systems and show the relationship between the new concept and overlarge sets of Kirkman triple systems (OLKTS(v)). Further, some recursive constructions for OLGKS are given, and some new existence results of OLKTS(v) with a special properties are obtained, which plays an important role in the discussion of large sets of Kirkman triple systems.

More large sets of resolvable MTS and resolvable DTS with odd orders

In this paper, we first give a method to construct large sets of resolvable Mendelsohn triple systems of order q+2, where q=6t+1 is a prime power. Then, using a computer, we find solutions for t@?T={35,38,46,47,48,51,56,60}. Furthermore, by a method we introduced, large sets of resolvable directed triple systems with the same orders are obtained too. Finally, by the tripling construction and product construction for LRMTSs and LRDTSs, and by new results for LR-designs, we obtain the existence of an LRMTS(v) and an LRDTS(v) for all v of the formv=(6t+3)@?m@?M(2.7^m+1)@?n@?N(2.13^n+1),where t@?T and M and N are finite multisets of non-negative integers. This provides more infinite classes for LRMTSs and LRDTSs with odd orders.

The spectrum for large set of disjoint incomplete Latin squares

Abstract An incomplete Latin square LS(n+a,a) is a Latin square of order n+a with a missing subsquare of order a. A large set of disjoint LS(n+a,a)s, denoted by LDILS(n+a,a), consists of n disjoint LS(n+a,a)s. About the existence of LDILS, Zhu, Wu, Chen and Ge have already obtained some results (see Wu and Zhu, Bull Inst. Combin. Appl., to appear.). In this paper, we introduce a kind of auxiliary design LSm(n) and, using it, completely solve the existence problem of LDILS. The conclusion is that for any positive integer n and any integer a, 0⩽a⩽n , there exists an LDILS(n+a,a) if and only if (n,a)≠(2,1) and (6,5).

Packings and coverings of λKv by k-circuits with one chord

Abstract Let λKv be the complete multigraph with v vertices, where any two distinct vertices x and y are joined by λ edges {x,y}. Let G be a finite simple graph. A G-packing design (G-covering design) of λKv, denoted by (v,G,λ)-PD ((v,G,λ)-CD) is a pair (X, B ) , where X is the vertex set of Kv and B is a collection of subgraphs of Kv, called blocks, such that each block is isomorphic to G and any two distinct vertices in Kv are joined in at most (at least) λ blocks of B . A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design has more (fewer) blocks. In this paper, the discussed graphs are Ck(r), i.e., one circle of length k with one chord, where r is the number of vertices between the ends of the chord, 1⩽r

Large sets of extended directed triple systems with even orders

For three types of triples: unordered, cyclic and transitive, the corresponding extended triple, extended triple system and their large sets are introduced. The existence of LESTS(@u) and LEMTS(@u) were completely solved. In this paper, we shall discuss the existence problem of LEDTS(@u) and give the following conclusion: there exists an LEDTS(@u) for any even @u except @u=4. The existence of LEDTS(@u) with odd order @u will be discussed in another paper, we are working at it.

Completing the spectrum for large sets of extended directed triple systems

Extended triple systems are firstly introduced by Johnson and Mendelson (1972). The existence of the large sets of extended directed triple systems L E D T S ( v ) has been basically solved except possibly five orders v = 95 , 143 , 167 , 203 , 215 , see Liu and Kang (2009, 2010). But, the existence proof of L E D T S ( 12 k + 11 ) is very complicated. In this paper, we give a simple existence proof and get the existence of L E D T S s for the remaining five orders. Finally, the spectrum for L E D T S s is completely determined.