Clique decompositions of multipartite graphs and completion of Latin squares

Barber, Ben; Kühn, Daniela; Lo, Allan; Osthus, Deryk; Taylor, A. M. Robert

doi:10.1016/j.jcta.2017.04.005

Cited by 25 publications

(51 citation statements)

References 21 publications

(25 reference statements)

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“…To see (2), note that G 3 is C 4 -divisible since it was obtained by removing edge-disjoint copies of C 4 from G. In particular, this means that G 3 is 2-divisible and so ∪ e A C B ( , ) G 3 is even. Since e G ( ) 3 is also even, the result follows. We use (2) to cover all remaining edges in…”

Section: Type 1 Extremalmentioning

confidence: 71%

“…The results in [2] and [8] can be combined with bounds for the fractional version of this problem in [1] and [7] to obtain good explicit bounds on the F -decomposition threshold. Corresponding results for the multipartite setting (with applications to the completion of partially filled Latin squares) were considered in [3], [4] and [13]. The only known exact minimum degree bound (before Theorem 1.2) was obtained by Yuster [16] who studied the case when F is a tree.…”

Section: Introductionmentioning

confidence: 99%

“…If the edges are disjoint, find paths of length ≥ k − 1 3 between their endpoints in A and in B. Again, properties (S1) to (S4) allow us to find these paths so that they are edge-disjoint and each vertex appears as an interior vertex on at most η n 3 . ■…”

mentioning

confidence: 99%

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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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Section: Type 1 Extremalmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

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On the exact decomposition threshold for even cycles

Taylor

2018

Journal of Graph Theory

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“…Related results on clique-decompositions of r-partite graphs have been obtained by Barber, Kühn, Lo, Osthus and Taylor [3]. (These can be combined with results on fractional decompositions of r-partite graphs by Dukes [9] and Montgomery [23].)…”

Section: Introductionmentioning

confidence: 81%

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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“…It was also proven by Chowla et al that there exist constants

C, α > 0

such that

N (n) \geq C n^{α}

for large enough

n

. This result has been improved in various ways over the years and recently Barber et al proved that one can prescribe a substantial number of entries in

r

partial Latin squares and still be able to extend these partial Latin squares to a set of

r

MOLS, where

r

grows with

n

.…”

Section: Introductionmentioning

confidence: 99%

Triples of orthogonal Latin and Youden rectangles for small orders

Jäger

Markström

Öhman

et al. 2018

J of Combinatorial Designs

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We have performed a complete enumeration of nonisotopic triples of mutually orthogonal k × n Latin rectangles for k ≤ n ≤ 7. Here we will present a census of such triples, classified by various properties, including the order of the autotopism group of the triple. As part of this, we have also achieved the first enumeration of pairwise orthogonal triples of Youden rectangles. We have also studied orthogonal triples of k × 8 rectangles which are formed by extending mutually orthogonal triples with nontrivial autotopisms one row at a time, and requiring that the autotopism group is nontrivial in each step. This class includes a triple coming from the projective plane of order 8. Here we find a remarkably symmetrical pair of triples of 4 × 8 rectangles, formed by juxtaposing two selected copies of complete sets of mutually orthogonal Latin squares of order 4.

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Clique decompositions of multipartite graphs and completion of Latin squares

Cited by 25 publications

References 21 publications

On the exact decomposition threshold for even cycles

On the exact decomposition threshold for even cycles

Edge-decompositions of graphs with high minimum degree

Triples of orthogonal Latin and Youden rectangles for small orders

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