On the decomposition threshold of a given graph

Glock, Stefan; Kühn, Daniela; Lo, Allan; Montgomery, Richard; Osthus, Deryk

doi:10.1016/j.jctb.2019.02.010

Cited by 32 publications

(54 citation statements)

References 24 publications

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“…Answering a question of Erdős , two sets of authors in and independently proved that for any

m \geq 2

ex false(n, {MJX-tex-caligraphicscriptF}_{3} (4, 2) \cup {MJX-tex-caligraphicscriptF}_{3} (5, 3) \cup ⋯ \cup {MJX-tex-caligraphicscriptF}_{3} (m + 2, m) false) = true(\frac{1}{3} + o (1) true) true(\frac{0.0pt}{n} true) .

It was suggested in [, Conjecture 7.2] that a similar statement must hold for all

r \geq 3, normala normaln normald m \geq 2

, namely

ex false(n, {MJX-tex-caligraphicscriptF}_{r} (r + 1, 2) \cup {MJX-tex-caligraphicscriptF}_{r} (r + 2, 3) \cup ⋯ \cup {MJX-tex-caligraphicscriptF}_{r} (r + m - 1, m) false) = true(\frac{1}{r} + o (1) true) true(\frac{0.0pt}{n} true) .

We will prove this statement for

m = 3

by constructing

F_{r} (r + 2, 3)

‐free (or, in terms of ,

3

‐sparse) partial Steiner

(r - 1, r, n)

‐systems with

false(\frac{1}{r} - o (1) false) true(\frac{0.0pt}{n} true)

blocks (edges). The case

r = 3

is trivial, because any

...

…”

Section: Introductionmentioning

confidence: 99%

Approximate Steiner (r − 1, r, n)‐systems without three blocks on r + 2 points

Sidorenko

2019

J of Combinatorial Designs

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For a family MJX-tex-caligraphicscriptF of r‐graphs, let ex(n,MJX-tex-caligraphicscriptF) denote the maximum number of edges in an MJX-tex-caligraphicscriptF‐free r‐graph on n vertices. Let Fr(v,e) denote the family of all r‐graphs with e edges and at most v vertices. We prove that ex(n,MJX-tex-caligraphicscriptFr(r+1,2)∪MJX-tex-caligraphicscriptFr(r+2,3))=false(1r−o(1)false)true(0.0ptnr−1true).

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“…Answering a question of Erdős , two sets of authors in and independently proved that for any

m \geq 2

ex false(n, {MJX-tex-caligraphicscriptF}_{3} (4, 2) \cup {MJX-tex-caligraphicscriptF}_{3} (5, 3) \cup ⋯ \cup {MJX-tex-caligraphicscriptF}_{3} (m + 2, m) false) = true(\frac{1}{3} + o (1) true) true(\frac{0.0pt}{n} true) .

It was suggested in [, Conjecture 7.2] that a similar statement must hold for all

r \geq 3, normala normaln normald m \geq 2

, namely

ex false(n, {MJX-tex-caligraphicscriptF}_{r} (r + 1, 2) \cup {MJX-tex-caligraphicscriptF}_{r} (r + 2, 3) \cup ⋯ \cup {MJX-tex-caligraphicscriptF}_{r} (r + m - 1, m) false) = true(\frac{1}{r} + o (1) true) true(\frac{0.0pt}{n} true) .

We will prove this statement for

m = 3

by constructing

F_{r} (r + 2, 3)

‐free (or, in terms of ,

3

‐sparse) partial Steiner

(r - 1, r, n)

‐systems with

false(\frac{1}{r} - o (1) false) true(\frac{0.0pt}{n} true)

blocks (edges). The case

r = 3

is trivial, because any

...

…”

Section: Introductionmentioning

confidence: 99%

Approximate Steiner (r − 1, r, n)‐systems without three blocks on r + 2 points

Sidorenko

2019

J of Combinatorial Designs

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“…The following result (taken from the more general statement for

F

‐decompositions, Lemma 5.1 in ) finds an approximate

C_{4}

‐decomposition of

G

leaving only a very small (and very restricted) leftover

H

.…”

Section: Cycles Of Length Fourmentioning

confidence: 82%

“…A

(δ, μ, m)

‐vortex respecting

(A, B)

G

is a sequence

U_{0} \supseteq U_{1} \supseteq \dots \supseteq U_{ℓ}

such that

U_{0} = V (G)

;

∣ U_{i} \cap X ∣ = ⌊ μ ∣ U_{i - 1} \cap X ∣ ⌋

for all

1 \leq i \leq ℓ

and each

X \in {A, B}

, and

∣ U_{ℓ} ∣ = m

;

d_{G} (x, U_{i} \cap X) \geq δ ∣ U_{i} \cap X ∣

, for all

1 \leq i \leq ℓ

, each

X \in {A, B}

and all

x \in U_{i - 1} \ X

. The following observation guarantees a vortex in

G

. It is proved by repeatedly applying the Chernoff bound given by Lemma (for more details, see the proof of Lemma 4.3 in ).…”

Section: Decompositions Of Bipartite Graphsmentioning

confidence: 94%

“…We require the following simple proposition on decompositions of cliques. It is a special case of Wilson s Theorem and is proved very easily (see , for example).…”

Section: Decompositions Of Bipartite Graphsmentioning

confidence: 99%

“…Let

G

be a graph on

n

vertices. A

(δ, μ, m)

‐vortex in

G

(as defined in ) is a sequence

U_{0} \supseteq U_{1} \supseteq \dots \supseteq U_{ℓ}

such that:

U_{0} = V (G)

;

∣ U_{i} ∣ = ⌊ μ ∣ U_{i - 1} ∣ ⌋

, for all

1 \leq i \leq ℓ

, and

∣ U_{ℓ} ∣ = m

;

d_{G} (x, U_{i}) \geq δ ∣ U_{i} ∣

, for all

1 \leq i \leq ℓ

and all

x \in U_{i - 1}

. We use Lemma 4.3 from to find a vortex in

G

.…”

Section: Cycles Of Length Fourmentioning

confidence: 99%

See 2 more Smart Citations

On the exact decomposition threshold for even cycles

Taylor

2018

Journal of Graph Theory

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A graph G has a C k‐decomposition if its edge set can be partitioned into cycles of length k. We show that if δ ( G ) ≥ 2 ∣ G ∣ ∕ 3 − 1, then G has a C 4‐decomposition, and if δ ( G ) ≥ ∣ G ∣ ∕ 2, then G has a C 2 k‐decomposition, where k ∈ double-struckN and k ≥ 4 (we assume G is large and satisfies necessary divisibility conditions). These minimum degree bounds are best possible and provide exact versions of asymptotic results obtained by Barber, Kühn, Lo and Osthus. In the process, we obtain asymptotic versions of these results when G is bipartite or satisfies certain expansion properties.

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Latin squares from multiplication tables

Dębski

Grytczuk

2021

J of Combinatorial Designs

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Consider the usual multiplication table n×n in which all entries strictly greater than n were deleted. Can we fill in the empty places so that the resulting table will become a multiplication table of a group? This intriguing question arose in connection to the famous Graham's Greatest Common Divisor Problem. We prove here a weaker statement, namely, that one may always complete such a partial multiplication table to a Latin square. Our result is more general, as it guarantees the possibility of completion of any partial Latin square, provided that the shape of the empty part satisfies certain conditions. We also point on connections of our result to a far‐reaching extension of Graham's problem involving graphs and homogeneous arithmetic progressions.

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On the decomposition threshold of a given graph

Abstract: We study the F -decomposition threshold δF for a given graph F . Here an F

Cited by 32 publications

References 24 publications

Approximate Steiner (r − 1, r, n)‐systems without three blocks on r + 2 points

Approximate Steiner (r − 1, r, n)‐systems without three blocks on r + 2 points

On the exact decomposition threshold for even cycles

Latin squares from multiplication tables

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