P 3-factorization of complete multipartite graphs

Ushio, Kazuhiko; Tsuruno, Reiji

doi:10.1007/bf01788695

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Introduction2

Lemma 421

By Induction Hypothesis1

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2015

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Cited by 13 publications

(4 citation statements)

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“…Necessary and suf®cient conditions for the existence of S k -decompositions of complete multipartite graphs are given in [15] (for equal size parts) and in [12] (for different size parts).…”

Section: Introductionmentioning

confidence: 99%

Star factorizations of graph products

2001

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Section: Introductionmentioning

confidence: 99%

Star factorizations of graph products

2001

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“…The case (0, 6, 0) follows by [37]. For the remaining cases, the groups are {1, 2 The case (2, 0, 6) corresponds to a (P 2 , P 4 ) − URD(12; 2, 6) which is known to exist [16].…”

Section: Lemma 42mentioning

confidence: 99%

“…Proof. The case (0, 6, 0) follows by [23]. For the remaining cases, take the groups to be {1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12} and the classes listed below: Proof.…”

Section: By Induction Hypothesismentioning

confidence: 99%

Uniformly resolvable decompositions of Kv into paths on two, three and four vertices

Faro

Milici

Tripodi

2015

Discrete Mathematics

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“…Ushio in [5] gave a necessary and sufficient condition for existence of P 3 -factorization of K m,n . Some further work was done by Ushio and Tsuruno in [7], Du in [1,2], and Wang and Du in [9]. In paper [6], Ushio proposed the following conjecture [6,Conjecture 5.3].…”

Section: Introductionmentioning

confidence: 95%

P5-factorization of complete bipartite graphs

Wang

Du²

2008

Discrete Mathematics

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A P k -factor of complete bipartite graph K m,n is a spanning subgraph of K m,n such that every component is a path of length k. A P k -factorization of K m,n is a set of edge-disjoint P k -factors of K m,n which is a partition of the set of edges of K m,n . When k is an even number, the spectrum problem for a P k -factorization of K m,n has been completely solved. When k is an odd number, Ushio in 1993 proposed a conjecture. However, up to now we only know that Ushio Conjecture is true for k = 3. In this paper we will show that Ushio Conjecture is true when k = 5. That is, we shall prove that a necessary and sufficient condition for the existence of a P 5 -factorization of K m,n is (1) 3n 2m, (2) 3m 2n, (3) m + n ≡ 0 (mod 5), and (4) 5mn/[4(m + n)] is an integer.

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P 3-factorization of complete multipartite graphs

Cited by 13 publications

References 3 publications

Star factorizations of graph products

Star factorizations of graph products

Uniformly resolvable decompositions of Kv into paths on two, three and four vertices

P5-factorization of complete bipartite graphs

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