Switching techniques for edge decompositions of graphs

Horsley, Daniel

doi:10.1017/9781108332699.006

Cited by 3 publications

(12 citation statements)

References 68 publications

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“…Proof Start with a (3, λ)-GDD of type n 3 c 1 . Then place a TS λ (n) on one group of size n. ✷ Lemma 4.6 There exists a ✸-ITS 2 (v; w, u, y; z 1 , z 2 ) for (v, w, u, y, z 1 , z 2 ) ∈ { (11,3,3,2,0,0), (11,4,3,2,1,0), (12,3,3,3,0,0), (13,3,3,4,0,0), (13,4,3,4,1,0), (14, 5, 4, 5, 2, 3), (18, 7, 4, 5, 2, 3), (22, 6, 6, 4, 0, 0), (23, 8, 7, 5, 2, 3), (23, 8, 7, 8, 2, 6), (24,8,6,6,2,0)}.…”

Section: Near the Boundmentioning

confidence: 99%

“…Proof By taking (n, c, λ) ∈ { (3,2,2), (3,3,2), (3,4,2), (6,4,2)}, apply Lemma 4.5 together with Theorem 1.1 and Lemma 4.2 to get the cases of (v, w, u, y, z 1 , z 2 ) ∈ { (11,3,3,2,0,0), (12,3,3,3,0,0), (13, 3, 3, 4, 0, 0), (22, 6, 6, 4, (1) a (3, λ)-IGDD of type (2m + l; m, l) 1 (3h; h, h) t−1 (2h; h, 0) s−t ,…”

Section: Near the Boundmentioning

confidence: 99%

“…Lemma 4.24 Let (w, u, x) ∈ {(6, 3, 1), (9,3,1), (9,6,1), (8,6,1), (11,9,1), (9,4,2), (10,9,2), (3,2,3), (6,4,3), (7,4,3), (8,4,3), (8,7,3), (9,8,3), (10,4,3), (10,7,3), (11,4,3), (11,10,3)…”

Section: λ =mentioning

confidence: 99%

“…For w ≡ 2 (mod 3) and u ≡ 0 (mod 3), if (w, u) ∈ { (8,6), (11,9)}, apply Lemma 4.24. If (w, u) ∈ { (5,3), (8,3), (11,3), (11,6)}, apply Lemma 4.25.…”

Section: λ =mentioning

confidence: 99%

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Triangle decompositions of $λK_v-λK_w-λK_u$

Li,

Chang,

Feng

2019

Preprint

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Denote by λK v the complete graph of order v with multiplicity λ. Let λK v − λK w − λK u be the graph obtained from λK v by the removal of the edges of two vertex disjoint complete multi-subgraphs with multiplicity λ of orders w and u, respectively. When λ is odd, it is shown that there exists a triangle decomposition of λK v −λK w −λK u if and only if v ≥ w+u+max{u, w},When λ is even, it is shown that for large enough v, the elementary necessary conditions for the existence of a triangle decomposition of λK v − λK w − λK u are also sufficient.

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Section: Near the Boundmentioning

confidence: 99%

Section: Near the Boundmentioning

confidence: 99%

Section: λ =mentioning

confidence: 99%

“…For w ≡ 2 (mod 3) and u ≡ 0 (mod 3), if (w, u) ∈ { (8,6), (11,9)}, apply Lemma 4.24. If (w, u) ∈ { (5,3), (8,3), (11,3), (11,6)}, apply Lemma 4.25.…”

Section: λ =mentioning

confidence: 99%

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Triangle decompositions of $λK_v-λK_w-λK_u$

Li,

Chang,

Feng

2019

Preprint

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“…This 'switching' method was first applied to packings of the complete graph [3][4][5], and has since been generalised to other graphs [6,12,14]. See [13] for a proof of Lemma 3 and a survey of switching techniques for graph decompositions. Lemma 3 ([13]).…”

Section: Notation and Proof Strategymentioning

confidence: 99%

DecomposingKu+w−Kuinto cycles of prescribed lengths

Horsley

Hoyte

2017

Discrete Mathematics

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a b s t r a c tWe prove that the complete graph with a hole K u+w − K u can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, each cycle has length at most min(u, w), and the longest cycle is at most three times as long as the second longest. This generalises existing results on decomposing the complete graph with a hole into cycles of uniform length, and complements work on decomposing complete graphs, complete multigraphs, and complete multipartite graphs into cycles of arbitrary specified lengths.

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Graphs Isomorphisms Under Edge-Replacements and the Family of Amoebas

Caro¹,

Hansberg

Montejano

2023

Electron. J. Combin.

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This paper offers a systematic study of a family of graphs called amoebas. Amoebas recently emerged from the study of forced patterns in $2$-colorings of the edges of the complete graph in the context of Ramsey-Turan theory and played an important role in extremal zero-sum problems. Amoebas are graphs %with a unique behavior with regards to defined by means of the following operation: Let $G$ be a graph and let $e\in E(G)$ and $e'\in E(\overline{G})$. If the graph $G'=G-e+e'$ is isomorphic to $G$, we say $G'$ is obtained from $G$ by performing a feasible edge-replacement. We call $G$ a local amoeba if, for any two copies $G_1$, $G_2$ of $G$ on the same vertex set, $G_1$ can be transformed into $G_2$ by a chain of feasible edge-replacements. On the other hand, $G$ is called global amoeba if there is an integer $t_0 \ge 0$ such that $G \cup tK_1$ is a local amoeba for all $t \ge t_0$. To model the dynamics of the feasible edge-replacements of $G$, we define a group $\rm{Fer}(G)$ that satisfies that $G$ is a local amoeba if and only if $\rm{Fer}(G) \cong S_n$, where $n$ is the order of $G$. Via this algebraic setting, a deeper understanding of the structure of amoebas and their intrinsic properties comes into light. Moreover, we present different constructions that prove the richness of these graph families showing, among other things, that any connected graph can be a connected component of a global amoeba, that global amoebas can be very dense and that they can have, in proportion to their order, large clique and chromatic numbers. Also, a family of global amoeba trees with a Fibonacci-like structure and with arbitrary large maximum degree is constructed.

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Switching techniques for edge decompositions of graphs

Cited by 3 publications

References 68 publications

Triangle decompositions of $λK_v-λK_w-λK_u$

Triangle decompositions of $λK_v-λK_w-λK_u$

DecomposingKu+w−Kuinto cycles of prescribed lengths

Graphs Isomorphisms Under Edge-Replacements and the Family of Amoebas

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