Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Pv-factorization of Km,n is a set of edge-disjoint Pv-factors of Km,n which partition the set of edges of Km,n. When v is an even number, Wang and Ushio gave a necessary and sufficient condition for existence of Pv-factorization of Km,n. When k is an odd number, Ushio in 1993 proposed a conjecture. Very recently, we have proved that Ushio’s conjecture is true when v = 4k − 1. In this paper we shall show that Ushio Conjecture is true when v = 4k +1, and then Ushio’s conjecture is true. That is, we will prove that a necessary and sufficient condition for the existence of a P4k+1-factorization of Km,n is (i) 2km ≤ (2k + 1)n,(ii) 2kn ≤ (2k +1)m, (iii) m+n ≡ 0 (mod 4k +1), (iv) (4k +1)mn/[4k(m+n)] is an integer.

We shall refer to a strong partially balanced design SPBD(v, b, k; λ, 0) whose b is the maximum number of blocks in all SPBD(v, b, k; λ, 0), as an optimal strong partially balanced design, briefly OSPBD(v, k, λ). The author inpaper (Discrete Math. 279 (2004) 173) investigated the existence of OSPBD(v, 5, 1) and gave the spectra of OSPBD(v, 5, 1) for v ≡ 0, 1, 3 (mod 4). Inthis article we shall investigate the existence of OSPBD(v, 5, 1) and give the spectrum of OSPBD(v, 5, 1) for the remaining case v ≡ 2 (mod 4).

We shall refer to a diagonal Latin square which is orthogonal to its (3, 2, 1)-conjugate, and the latter is also a diagonal Latin square, as a (3, 2, 1)-conjugate orthogonal diagonal Latin square, briefly CODLS. This article investigates the spectrum of CODLS with a missing sub-square. The main purpose of this paper is two-fold. First of all, we show that for any positive integers n ≥ 1, a CODLS of order v with a missing subsquare of order n exists if v ≥ 13n/4 + 93 and v − n is even. Secondly, we show that for 2 ≤ n ≤ 6, a CODLS of order v with a missing subsquare of order n exists if and only if v ≥ 3n+2 and v − n is even, with one possible exception.

Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Kp,q factorizeation of Km,n is a set of edge-disjoint Kp,q-factors of Km,n which partition the set of edges of Km,n. When p=1 and q is a prime number, Wang, in his paper “On K1,k-factorizations of a complete bipartite graph” (Discrete Math, 1994, 126: 359-364), investigated the K1,q-factorization of Km,n and gave a sufficient condition for such a factorizeation to exist. In the paper “K1,k-factorizations of complete bipartite graphs” (Discrete Math, 2002, 259: 301-306), Du and Wang extended Wang’s result to the case that q is any positive integer. In this paper, we give a sufficient condition for Km,n to have a Kp,q-factorization. As a special case, it is shown that the Martin’s BAC conjecture is true when p:q=k:(k+1) for any positive integer k.

A group divisible design GD (k, λ, t; tu ) isα-resolvable if its blocks can be partitioned into classes such that each point of the design occurs in precisely α blocks in each class. The necessary conditions for the existence of such a design are λt( u – 1) = r ( k – 1) , bk=rtu; k│αtu andα│ r. It is shown in this paper that these conditions are also sufficient when k =3, with some definite exceptions.