Design of a new balanced file organization scheme with the least redundancy

2007

Let λKm,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A Pv-factorization of λKm,n is a set of edge-disjoint Pv-factors of λKm,n which partition the set of edges of λKm,n. When v is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a Pv-factorization of λKm,n. When v is an odd number, we have proposed a conjecture. Very recently, we have proved that the conjecture is true when v = 4k − 1. In this paper we shall show that the conjecture is true when v = 4k + 1, and then the conjecture is true. That is, we will prove that the necessary and sufficient conditions for the existence of a P 4k+1 -factorization of λKm,n are (1) 2km (2k +1)n, (2) 2kn (2k +1)m, (3) m + n ≡ 0 (mod 4k + 1), (4) λ(4k + 1)mn/[4k(m + n)] is an integer.

Section: Introductionmentioning

confidence: 99%

The spectrum of path factorization of bipartite multigraphs

2007

“…Especially, Yamamoto et al [3] have given some applications in HU BF S 2 schemes of database systems. There are also some known results on the existence of the P v -factorizations of λK m,n .…”

Section: Introductionmentioning

confidence: 99%

P 4k−1-factorization of bipartite multigraphs

2006

Let λKm,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A Pv-factorization of λKm,n is a set of edge-disjoint Pv-factors of λKm,n which partition the set of edges of λKm,n. When v is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a Pv-factorization of λKm,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true for v = 3. In this paper we will show that the conjecture is true when v = 4k − 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P 4k−1 -factorization of λKm,n is (1) (2k − 1)m 2kn, (2) (2k − 1)n 2km, (3) m + n ≡ 0 (mod 4k − 1), (4) λ(4k − 1)mn/[2(2k − 1)(m + n)] is an integer.

“…Especially, Yamamoto et al [3,4] have given some applications in HU BF S 2 and HU BM F S 2 schemes of database systems. There are also some known results on the existence of the P v -factorizations and K p,q -factorizations of K m,n .…”

Section: Introductionmentioning

confidence: 99%

The proof of Ushio’s conjecture concerning path factorization of complete bipartite graphs

2006

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 P 4k+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.