P3-Factorization of complete bipartite graphs

Ushio, Kazuhiko

doi:10.1016/0012-365x(88)90227-0

Cited by 36 publications

(10 citation statements)

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“…The existence of the P 3 -factorization of λK m,n has been completely solved by Ushio [6] , and Wang and Du [7] . In [1], Ushio proposed the following conjecture for the case λ = 1 (Conjecture 5.3 in [1]).…”

Section: Introductionmentioning

confidence: 99%

The spectrum of path factorization of bipartite multigraphs

Wang

2007

SCI CHINA SER A

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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.

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Section: Introductionmentioning

confidence: 99%

The spectrum of path factorization of bipartite multigraphs

Wang

2007

SCI CHINA SER A

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“…The existence of the P 3 -factorization of λK m,n has been completely solved by Ushio [6] , and Wang and Du [7] . In ref.…”

Section: Introductionmentioning

confidence: 99%

P 4k−1-factorization of bipartite multigraphs

Wang

2006

SCI CHINA SER A

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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.

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“…Let the short path be (1,2). The one long path from each block of A1 form one resolution class of the desired design.…”

Section: W 1 Introductionmentioning

confidence: 99%

“…Do the same work for A2, but let the letter short path be (2,3). (2) A P3-factorization of K~ is given below:…”

Section: W 1 Introductionmentioning

confidence: 99%

P 3-factorization of complete multipartite graphs

1999

Appl. Math. Chin. Univ.

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In this note it is shown that a necessary and sufficient condition for the existence of a Pa-factorization of complete multipartite graph aK~, is (1) rn~3, (2) mn~-O(mod 3) and (3) 2(m--1)n~0(mod 4). w 1 IntroductionLet P3 be a path on 3-points and K~, be a complete multipartite graph. AK~ is the multigraph obtained from K~, by replacing each edge e of K~ by a set of a edges all having the same end vertices as e. A subgraph F of ).KT, is called a spanning subgraph of AK~, if F contains all the vertices of AKT~. A spanning subgraph F of AK~ is called a Pa-factor if each component of F is isomorphic to Pa. If AK:, is expressed as a line-disjoint sum of P3-factors, then this sum is called a Pa-factorization of 2K:,.The spectrum problems for P3-factorization of complete graph AK~ and of complete bipartite graph Kin., have been completely solved by [-1,2]. In this note, a necessary and sufficient condition for the existence of a Pa-factorization of complete multipartite grap.h AKT, will be given.For this purpose we need a group divisible design. A group divisible design GDD (m, n, 3,2) consists of a collection of n-subsets, called groups, of a ran-set X and a collection of 3-subsets, called blocks, such that (1) the groups form a partition of X,(2) each pair of elements from different groups occurs together in exactly ), blocks,(3) no block contains two elements from the same group.A GDD is said to be resolvable if its block can be partitioned into r parallel classes, each of which partitions the set X. We denote it by RGDD. From simple counting we

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P3-Factorization of complete bipartite graphs

Cited by 36 publications

References 1 publication

The spectrum of path factorization of bipartite multigraphs

The spectrum of path factorization of bipartite multigraphs

P 4k−1-factorization of bipartite multigraphs

P 3-factorization of complete multipartite graphs

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