Fractional clique decompositions of dense graphs and hypergraphs

Barber, Ben; Kühn, Daniela; Lo, Allan; Montgomery, Richard; Osthus, Deryk

doi:10.1016/j.jctb.2017.05.005

Cited by 24 publications

(76 citation statements)

References 24 publications

(40 reference statements)

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“…Our argument here and that in [2] is purely combinatorial. In particular, the proofs of Theorems 4.7 and 6.3 together yield a combinatorial proof of Wilson's theorem [30,31,32,33] that every large F -divisible clique has an F -decomposition.…”

Section: Introductionmentioning

confidence: 54%

“…is given in [7,8] for small values of r and in [2] for large values of r. Theorem 4.6 (Dukes [7,8]). For r ∈ N with r ≥ 2, δ *…”

Section: Fractional and Approximate F -Decompositionsmentioning

confidence: 98%

“…To obtain the explicit bound on δ(G), we apply results of Dukes [7,8] as well as Barber, Kühn, Lo, Montgomery and Osthus [2] on fractional decompositions in graphs of large minimum degree together with a result of Haxell and Rödl [16] relating fractional decompositions to approximate decompositions. We collect these tools in Section 4.…”

Section: Sketches Of Proofsmentioning

confidence: 99%

“…A result of Yuster [38] implies that we can consider the fractional K χ(F ) -decomposition threshold instead of the fractional F -decomposition threshold. We will then use the results from [7,8] as well as from [2], which guarantee a fractional K r -decomposition of any graph on n vertices with minimum degree at least (1 − 2/(9r 2 (r − 1) 2 ))n and minimum degree at least (1 − 1/10 4 r 3/2 )n, respectively. Any improvement in the fractional K r -decomposition threshold would immediately imply better bounds in Theorem 1.2 (see Theorem 6.3).…”

Section: Introductionmentioning

confidence: 99%

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Edge-decompositions of graphs with high minimum degree

Barber

Kühn

et al. 2015

Electronic Notes in Discrete Mathematics

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Abstract. A fundamental theorem of Wilson states that, for every graph F , every sufficiently large F -divisible clique has an F -decomposition. Here a graph G is F -divisible if e(F ) divides e(G) and the greatest common divisor of the degrees of F divides the greatest common divisor of the degrees of G, and G has an F -decomposition if the edges of G can be covered by edge-disjoint copies of F . We extend this result to graphs G which are allowed to be far from complete. In particular, together with a result of Dross, our results imply that every sufficiently large K3-divisible graph of minimum degree at least 9n/10 + o(n) has a K3-decomposition. This significantly improves previous results towards the long-standing conjecture of Nash-Williams that every sufficiently large K3-divisible graph with minimum degree at least 3n/4 has a K3-decomposition. We also obtain the asymptotically correct minimum degree thresholds of 2n/3 + o(n) for the existence of a C4-decomposition, and of n/2 + o(n) for the existence of a C 2 -decomposition, where ≥ 3. Our main contribution is a general 'iterative absorption' method which turns an approximate or fractional decomposition into an exact one. In particular, our results imply that in order to prove an asymptotic version of Nash-Williams' conjecture, it suffices to show that every K3-divisible graph with minimum degree at least 3n/4 + o(n) has an approximate K3-decomposition.

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Section: Introductionmentioning

confidence: 54%

“…is given in [7,8] for small values of r and in [2] for large values of r. Theorem 4.6 (Dukes [7,8]). For r ∈ N with r ≥ 2, δ *…”

Section: Fractional and Approximate F -Decompositionsmentioning

confidence: 98%

Section: Sketches Of Proofsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Edge-decompositions of graphs with high minimum degree

Barber

Kühn

et al. 2015

Electronic Notes in Discrete Mathematics

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“…For each

r \geq 4

, Yuster showed that

δ_{K_{r}}^{*} \leq 1 - 1 / (9 r^{10})

and gave a construction showing that

δ_{K_{r}}^{*} \geq (1 - 1 / (r + 1)) n

. Dukes used tools from linear algebra to show that

δ_{K_{r}}^{*} \leq 1 - 2 / (9 r^{2} {(r - 1)}^{2})

, before Barber, Kühn, Lo, Osthus, and the current author were able to generalize and extend Dross's methods for fractional triangle decompositions to show that

δ_{K_{r}}^{*} \leq 1 - 1 / (10^{4} r^{3 / 2})

. In this paper, we show that, for each

r \geq 4, δ_{K_{r}}^{*} \leq 1 - 1 / (100 r)

.…”

Section: Introductionmentioning

confidence: 91%

Fractional clique decompositions of dense graphs

Montgomery

2018

Random Struct Algorithms

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For each r≥4, we show that any graph G with minimum degree at least (1−1/(100r))|G| has a fractional Kr‐decomposition. This improves the best previous bounds on the minimum degree required to guarantee a fractional Kr‐decomposition given by Dukes (for small r) and Barber, Kühn, Lo, Montgomery, and Osthus (for large r), giving the first bound that is tight up to the constant multiple of r (seen, for example, by considering Turán graphs). In combination with work by Glock, Kühn, Lo, Montgomery, and Osthus, this shows that, for any graph F with chromatic number χ(F)≥4, and any ε>0, any sufficiently large graph G with minimum degree at least (1−1/(100χ(F))+ε)|G| has, subject to some further simple necessary divisibility conditions, an (exact) F‐decomposition.

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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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Fractional clique decompositions of dense graphs and hypergraphs

Cited by 24 publications

References 24 publications

Edge-decompositions of graphs with high minimum degree

Edge-decompositions of graphs with high minimum degree

Fractional clique decompositions of dense graphs

On the exact decomposition threshold for even cycles

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