C 4p -frame of complete multipartite multigraphs

Chitra, V.; Vadivu, A. Shanmuga; Muthusamy, Appu

doi:10.1007/s00010-012-0167-5

Cited by 5 publications

(6 citation statements)

References 17 publications

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“…The existence of

(k, λ)

‐cycle frame of

(K_{m} \otimes I_{n}) (λ)

, where

k \in {3, 4, 5, 6, 2 p, 4 p ∣ p is a prime integer}

have been completely solved in . In this paper, the existence of

(2 k, λ)

‐cycle frame of

(K_{m} \otimes I_{n}) (λ)

k \geq 2

is an integer, has been settled completely.…”

Section: Resultsmentioning

confidence: 92%

“…Chitra et al. proved that there exists a

(4 p, λ)

‐cycle frame of

(K_{m} \otimes I_{n}) (λ)

if and only if

n (m - 1) \equiv 0 (mod 4 p)

λ n \equiv 0 (mod 2)

m \geq 3

and obtained some necessary or sufficient conditions for the existence of a

(2 k, λ)

‐cycle frame of

(K_{m} \otimes I_{n}) (λ)

. Recently, Shanmuga Vadivu and Muthusamy proved that there exists a

({4, 6}, λ)

‐cycle frame of

(K_{m} \otimes I_{n}) (λ)

if and only if

λ n \equiv 0 (mod 2)

m \geq 3

(n, m) \notin {(1, 3), (2 t + 1, 2 s) ∣ t \geq 0, s \geq 2}

and also for

m \geq 3

and a prime p , there exists a

(2 p, λ)

‐cycle frame of

(K_{m} \otimes I_{n}) (λ)

if and only if

(m - 1) n \equiv 0 (mod 2 p)

λ n \equiv 0 (mod 2)

, and

…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Cycle Frames of Complete Multipartite Multigraphs - III

Muthusamy

Vadivu

2013

J. Combin. Designs

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For two graphs G and H their wreath product G ⊗ H has vertex set V (G) × V (H)in which two vertices (g 1 , h 1 ) and (g 2 , h 2 ) are adjacent whenever g 1 g 2 ∈ E(G) or g 1 = g 2 and h 1 h 2 ∈ E(H). Clearly, K m ⊗ I n , where I n is an independent set on n vertices, is isomorphic to the complete m-partite graph in which each partite set has exactly n vertices. A 2-regular subgraph of the complete multipartite graph K m ⊗ I n containing vertices of all but one partite set is called partial 2-factor. For an integer λ, G(λ) denotes a graph G with uniform edge multiplicity λ. Let J be a set of integers. If (K m ⊗ I n )(λ) can be partitioned into edge-disjoint partial 2-factors consisting cycles of lengths from J, then we say that (K m ⊗ I n )(λ) has a ( J, λ)-cycle frame. In this paper, we show that for m ≥ 3 and k ≥ 2, there exists a (2k, λ)-cycle frame of (K m ⊗ I n )(λ) if and only if (m − 1)n ≡ 0 (mod 2k) and λn ≡ 0 (mod 2). In fact our results completely solve the existence of a (2k, λ)-cycle frame of (K m ⊗ I n )(λ). C 2013 Wiley Periodicals, Inc. J. Combin.Designs 00: [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] 2013

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“…The existence of

(k, λ)

‐cycle frame of

(K_{m} \otimes I_{n}) (λ)

, where

k \in {3, 4, 5, 6, 2 p, 4 p ∣ p is a prime integer}

have been completely solved in . In this paper, the existence of

(2 k, λ)

‐cycle frame of

(K_{m} \otimes I_{n}) (λ)

k \geq 2

is an integer, has been settled completely.…”

Section: Resultsmentioning

confidence: 92%

“…Chitra et al. proved that there exists a

(4 p, λ)

‐cycle frame of

(K_{m} \otimes I_{n}) (λ)

if and only if

n (m - 1) \equiv 0 (mod 4 p)

λ n \equiv 0 (mod 2)

m \geq 3

and obtained some necessary or sufficient conditions for the existence of a

(2 k, λ)

‐cycle frame of

(K_{m} \otimes I_{n}) (λ)

. Recently, Shanmuga Vadivu and Muthusamy proved that there exists a

({4, 6}, λ)

‐cycle frame of

(K_{m} \otimes I_{n}) (λ)

if and only if

λ n \equiv 0 (mod 2)

m \geq 3

(n, m) \notin {(1, 3), (2 t + 1, 2 s) ∣ t \geq 0, s \geq 2}

and also for

m \geq 3

and a prime p , there exists a

(2 p, λ)

‐cycle frame of

(K_{m} \otimes I_{n}) (λ)

if and only if

(m - 1) n \equiv 0 (mod 2 p)

λ n \equiv 0 (mod 2)

, and

…”

Section: Introductionmentioning

confidence: 99%

Cycle Frames of Complete Multipartite Multigraphs - III

Muthusamy

Vadivu

2013

J. Combin. Designs

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“…Another paper, published by Niu and Cao [17], settled the case where cycles of both length 3 and length 5 are allowed in the frame. The only result in the literature addressing cycles of larger lengths appears to be that of Chitra, Vadivu and Muthusamy [5], who solved the problem for l = 4p where p is a prime. In this paper we settle the existence of cycle frames of type n p for the longest possible cycle length (see Theorem 4.1).…”

Section: Introductionmentioning

confidence: 97%

Fair holey hamiltonian decompositions of complete multipartite graphs and long cycle frames

Erzurumluoğlu

Rodger

2015

Discrete Mathematics

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a b s t r a c tA k-factor of a graph G = (V (G), E(G)) is a k-regular spanning subgraph of G. A k-factorization is a partition of E(G) into k-factors. Let K (n, p) be the complete multipartite graph with p parts, each of size n. If V 1 , . . . , V p are the p parts of V (K (n, p)), then a holey k-factor of deficiencyHence a holey k-factorization is a set of holey k-factors whose edges partition E(K (n, p)). A holey hamiltonian decomposition is a holey 2-factorization of K (n, p) where each holey 2-factor is a connected subgraph of K (n, p) − V i for some i satisfying 1 ≤ i ≤ p. A (holey) k-factorization of K (n, p) is said to be fair if the edges between each pair of parts are shared as evenly as possible among the permitted (holey) factors. In this paper the existence of fair holey hamiltonian decompositions of K (n, p) is completely settled. This result simultaneously settles the existence of cycle frames of type n p for cycles of the longest length, being a companion for results in the literature for frames with short cycle length.

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“…M. Niu and H. Cao [23] proved that there exits a {3, 5} -ARCS of (K u ⊗ Kg )(λ) . Lot of results have been published on ARCS of (K u ⊗ Kg )(λ), see [5,7,9,22]. RCS of (K u × K g )(λ) has been studied by many authors [1,3,4,11].…”

Section: Introductionmentioning

confidence: 99%