Haitao Cao

Super-simple balanced incomplete block designs with block size 4 and index 5

In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super-simple designs are also useful in other constructions, such as superimposed codes and perfect hash families etc. The existence of super-simple (v,4,@l)-BIBDs have been determined for @l=2,3,4 and 6. When @l=5, the necessary conditions of such a design are that v=1,4(mod12) and v>=13. In this paper, we show that there exists a super-simple (v,4,5)-BIBD for each v=1,4(mod12) and v>=13.

On the existence of cycle frames and almost resolvable cycle systems

Suppose H is a complete m -partite graph K m ( n 1 , n 2 , ? , n m ) with vertex set V and m independent sets G 1 , G 2 , ? , G m of n 1 , n 2 , ? , n m vertices respectively. Let G = { G 1 , G 2 , ? , G m } . If the edges of λ H can be partitioned into a set C of k -cycles, then ( V , G , C ) is called a k -cycle group divisible design with index λ , denoted by ( k , λ ) -CGDD. A ( k , λ ) -cycle frame is a ( k , λ ) -CGDD ( V , G , C ) in which C can be partitioned into holey 2-factors, each holey 2-factor being a partition of V ? G i for some G i ? G . Stinson et al. have resolved the existence of ( 3 , λ ) -cycle frames of type g u . In this paper, we show that there exists a ( k , λ ) -cycle frame of type g u for k ? { 4 , 5 , 6 } if and only if g ( u - 1 ) ? 0 ( mod k ) , λ g ? 0 ( mod 2 ) , u ? 3 when k ? { 4 , 6 } , u ? 4 when k = 5 , and ( k , λ , g , u ) ? ( 6 , 1 , 6 , 3 ) . A k -cycle system of order n whose cycle set can be partitioned into ( n - 1 ) / 2 almost parallel classes and a half-parallel class is called an almost resolvable k -cycle system, denoted by k -ARCS ( n ) . Lindner et al. have considered the general existence problem of k -ARCS ( n ) from the commutative quasigroup for k ? 0 ( mod 2 ) . In this paper, we give a recursive construction by using cycle frames which can also be applied to construct k -ARCS ( n ) s when k ? 1 ( mod 2 ) . We also update the known results and prove that for k ? { 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 14 } there exists a k -ARCS ( 2 k t + 1 ) for each positive integer t with three known exceptions and four additional possible exceptions.

Note: The existence of HGDDs with block size four and its application to double frames

In this note, we complete the existence spectra of HGDDs with block size four as well as double frames with block size three.

More results on cycle frames and almost resolvable cycle systems

Abstract Let J be a set of positive integers. Suppose m > 1 and H is a complete m -partite graph with vertex set V and m groups G 1 , G 2 , … , G m . Let | V | = v and G = { G 1 , G 2 , … , G m } . If the edges of λ H can be partitioned into a set C of cycles with lengths from J , then ( V , G , C ) is called a cycle group divisible design with index λ and order v , denoted by ( J , λ ) -CGDD. A ( J , λ ) -cycle frame is a ( J , λ ) -CGDD ( V , G , C ) in which C can be partitioned into holey 2-factors, each holey 2-factor being a partition of V ∖ G i into cycles for some G i ∈ G . The existence of ( k , λ ) -cycle frames of type g u with 3 ≤ k ≤ 6 has been solved completely. In this paper, we show that there exists a ( { 3 , 5 } , λ ) -cycle frame of type g u for any u ≥ 4 , λ g ≡ 0 ( mod 2 ) , ( g , u ) ≠ ( 1 , 5 ) , ( 1 , 8 ) and ( g , u , λ ) ≠ ( 2 , 5 , 1 ) . A k -cycle system of order n whose cycle set can be partitioned into ( n − 1 ) / 2 almost parallel classes and a half-parallel class is called an almost resolvable k -cycle system, denoted by k -ARCS ( n ) . It has been proved that for k ∈ { 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 14 } there exists a k -ARCS ( 2 k t + 1 ) for each positive integer t with three exceptions and four possible exceptions. In this paper, we shall show that there exists a k -ARCS ( 2 k t + 1 ) for all t ≥ 1 , 11 ≤ k ≤ 49 , k ≡ 1 ( mod 2 ) and t ≠ 2 , 3 , 5 .

Super-simple group divisible designs with block size 4 and index 5

Abstract In this paper, we investigate the existence of a super-simple (4, 5)-GDD of type g u and show that such a design exists if and only if u ≥ 4 , g ( u − 2 ) ≥ 10 , g ( u − 1 ) ≡ 0 ( mod 3 ) and u ( u − 1 ) g 2 ≡ 0 ( mod 12 ) .

A note on the Hamilton–Waterloo problem with C8-factors and Cm-factors

Abstract In this paper, we almost completely solve the Hamilton–Waterloo problem with C 8 -factors and C m -factors where the number of vertices is a multiple of 8 m .

Completing the spectrum of almost resolvable cycle systems with odd cycle length

In this paper, we construct almost resolvable cycle systems of order

Constructions of large sets of disjoint group-divisible designs LS ( 2 n 4 1 ) using a generalization of * LS ( 2 n )

4k+1

Constructions and bounds for separating hash families

for odd

Uniformly resolvable decompositions of K v into K 2 and K 1 , 3 graphs

k\ge 11