Enclosings of λ-fold 4-cycle systems

Newman, Nicholas; Rodger, C. A.

doi:10.1007/s10623-009-9353-4

Cited by 8 publications

(7 citation statements)

References 16 publications

(13 reference statements)

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“…Many papers have investigated the λ‐fold

3 – CS

enclosing problem (see ). The difficulty of this problem is clear since as yet there is no complete solution to this problem.…”

Section: Introductionmentioning

confidence: 97%

“…The difficulty of this problem is clear since as yet there is no complete solution to this problem. Newman and Rodger completely solved the problem of enclosing a λ‐fold

4 – CS

of order V into a

(λ + m)

‐fold

4 – CS

of order

v + u

for all

u, m \geq 1

in . In this paper, we study the k ‐cycle enclosing problem for λ‐fold

5 – CS

s settling the following conjecture in the difficult case where u = 1 (see Theorem ).…”

Section: Introductionmentioning

confidence: 99%

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Enclosings of λ‐Fold 5‐Cycle Systems: Adding One Vertex

Asplund

Rodger

Keranen

2013

J of Combinatorial Designs

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A k‐cycle system of a multigraph G is an ordered pair (V, C) where V is the vertex set of G and C is a set of k‐cycles, the edges of which partition the edges of G. A k‐cycle system of λKv is known as a λ‐fold k‐cycle system of order V. A k‐cycle system of λKv (V, C) is said to be enclosed in a k‐cycle system of (λ+m)Kv+u (V∪U,P) if C⊂P and u,m≥1. We settle the difficult enclosing problem for λ‐fold 5‐cycle systems with u = 1.

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“…Many papers have investigated the λ‐fold

3 – CS

enclosing problem (see ). The difficulty of this problem is clear since as yet there is no complete solution to this problem.…”

Section: Introductionmentioning

confidence: 97%

“…The difficulty of this problem is clear since as yet there is no complete solution to this problem. Newman and Rodger completely solved the problem of enclosing a λ‐fold

4 – CS

of order V into a

(λ + m)

‐fold

4 – CS

of order

v + u

for all

u, m \geq 1

in . In this paper, we study the k ‐cycle enclosing problem for λ‐fold

5 – CS

s settling the following conjecture in the difficult case where u = 1 (see Theorem ).…”

Section: Introductionmentioning

confidence: 99%

Enclosings of λ‐Fold 5‐Cycle Systems: Adding One Vertex

Asplund

Rodger

Keranen

2013

J of Combinatorial Designs

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“…There are a number of enclosing results in the case where λ < µ and the target decomposition consists of color classes each isomorphic to a cycle of one prescribed length [1,3,4,9,14,15,19,20] or each isomorphic to a cycle of one of a number of prescribed lengths [2,13]. However, less is known in the case where λ < µ and the target decomposition consists of spanning subgraphs.…”

Section: Introductionmentioning

confidence: 99%

Enclosings of decompositions of complete multigraphs in 2-edge-connected r-factorizations

Asplund

Charbit

Feghali

2019

Discrete Mathematics

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A decomposition of a multigraph G is a partition of its edges into subgraphs G(1), . . . , G(k). It is called an r-factorization if every G(i) is r-regular and spanning.If G is a subgraph of H, a decomposition of G is said to be enclosed in a decomposition of H if, for every 1 ≤ i ≤ k, G(i) is a subgraph of H(i).Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of λK n to be enclosed in some 2-edge-connected r-factorization of µK m for some range of values for the parameters n, m, λ, µ, r: r = 2, µ > λ and either m ≥ 2n − 1, or m = 2n − 2 and µ = 2 and λ = 1, or n = 3 and m = 4. We generalize their result to every r ≥ 2 and m ≥ 2n − 2. We also give some sufficient conditions for enclosing a given decomposition of λK n in some 2-edge-connected r-factorization of µK m for every r ≥ 3 and m = (2 − C)n, where C is a constant that depends only on r, λ and µ.

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“…We remark that, in the situation where the multiplicities of the two graphs are not equal, past work has focused on decompositions in which each subgraph of the target decomposition is isomorphic to a cycle of one fixed length or where each subgraph of the target decomposition is isomorphic to a cycle of one of a number of fixed lengths .…”

Section: Introductionmentioning

confidence: 99%

Enclosings of decompositions of complete multigraphs in 2‐factorizations

Feghali

Johnson

2018

J of Combinatorial Designs

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Let k, m, n, λ, and μ be positive integers. A decomposition of λKn into edge‐disjoint subgraphs G1,…,Gk is said to be enclosed by a decomposition of μKm into edge‐disjoint subgraphs H1,…,Hk if μ>λ and, after a suitable labeling of the vertices in both graphs, λKn is a subgraph of μKm and Gi is a subgraph of Hi for all i=1,⋯,k. In this paper, we continue the study of enclosings of given decompositions by decompositions that consist of spanning subgraphs. A decomposition of a graph is a 2‐factorization if each subgraph is 2‐regular and spanning, and is Hamiltonian if each subgraph is a Hamiltonian cycle. We give necessary and sufficient conditions for the existence of a 2‐factorization of μKn+m that encloses a given decomposition of λKn whenever μ>λ and m≥n−2. We also give necessary and sufficient conditions for the existence of a Hamiltonian decomposition of μKn+m that encloses a given decomposition of λKn whenever μ>λ and either m≥n−1 or n=3 and m=1, or μ=2, λ=1, and m=n−2.

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Enclosings of λ-fold 4-cycle systems

Cited by 8 publications

References 16 publications

Enclosings of λ‐Fold 5‐Cycle Systems: Adding One Vertex

Enclosings of λ‐Fold 5‐Cycle Systems: Adding One Vertex

Enclosings of decompositions of complete multigraphs in 2-edge-connected r-factorizations

Enclosings of decompositions of complete multigraphs in 2‐factorizations

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