Decomposing the vertex set of a hypercube into isomorphic subgraphs

Gruslys, Vytautas

doi:10.48550/arxiv.1611.02021

Cited by 5 publications

(14 citation statements)

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“…existence of a perfect H-packing in Q n : H has to be a subgraph of Q n ; and the order of H has to be a power of 2. Gruslys [5] showed that these two conditions are sufficient for large n, thus confirming a conjecture of Offner [12]. In fact, he showed that if H is an induced subgraph of Q k for some k and |H| is a power of 2, then there is a perfect packing of G into induced copies of H.…”

mentioning

confidence: 70%

“…The task of finding an l-partition is quite simple. In fact, it follows directly from the analogous result in [5] and the fact that (P 2l ) n can be partitioned into copies of Q n . For the sake of completeness, we include the proof here.…”

Section: Lower Bound On the Number Of Uncovered Verticesmentioning

confidence: 82%

“…Our proof follows the footsteps of Gruslys [5] who proved that if H is an induced subgraph of a hypercube whose order is a power of 2, then for large n there is a perfect packing of Q n into induced copies of H.…”

Section: Partitioning (P 2l ) Nmentioning

confidence: 93%

“…The existence of an |H|-partition of (P 2|H| ) n into induced copies of H is a simple observation (see Observation 6). The existence of a (1 mod |H|)-partition of (P 2|H| ) n into copies of H is more difficult to prove, but it is quite straightforward to adapt the methods of Gruslys [5] to work in our setting. These two facts, together the aforementioned result [7], form the proof of Theorem 3.…”

Section: Partitioning (P 2l ) Nmentioning

confidence: 99%

“…Unlike Observation 6, we cannot directly apply the analogous result of Gruslys [5]. Instead, we adapt his method to our setting.…”

Section: Lower Bound On the Number Of Uncovered Verticesmentioning

confidence: 99%

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Partitioning the Boolean lattice into copies of a poset

Gruslys

Leader

Tomon

2019

Journal of Combinatorial Theory, Series A

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Let H be an induced subgraph of the hypercube Q k , for some k. We show that for some c = c(H), the vertices of Q n can be partitioned into induced copies of H and a remainder of at most O(n c ) vertices. We also show that the error term cannot be replaced by anything smaller than log n. introductionGiven graphs G and H, an H-packing of G is a collection of vertex-disjoint copies of H in G. A perfect H-packing (also known as an H-factor) is an H-packing that covers all the vertices of the ground graph G (so, in order for G to have a perfect H-packing, |H| must divide |G|). A natural question asks for conditions on G that imply the existence of an H-factor. For example, a well researched question asks for the smallest minimum degree that implies the existence of an H-factor.If H is an edge (and more generally if H is a path), then, by Dirac's theorem [4], if G has n vertices and minimum degree at least n/2 (and |H| divides |G|), then G has a perfect H-packing. Corrádi and Hajnal [3] showed that δ(G) ≥ 2n/3 guarantees the existence of a perfect K 3 -packing and Hajnal and Szemerédi [8] extended this result by showing that if δ(G) ≥ (1 − 1/r)n then G has a perfect K r -packing. We remark that these conditions on δ(G) are best possible.After a series of papers by Alon and Yuster [1,2] and by Komlós, Sárközy and Szemerédi [9], Kuhn and Osthus [10] found the smallest minimum degree condition that guarantees the existence of an H-factor, up to an additive constant error term, and for all H.We consider a different problem, where instead of looking for H-packings in graphs of large minimum degree, we focus on H-packings of the hypercube Q n . There are two obvious conditions for the

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mentioning

confidence: 70%

Section: Lower Bound On the Number Of Uncovered Verticesmentioning

confidence: 82%

Section: Partitioning (P 2l ) Nmentioning

confidence: 93%

Section: Partitioning (P 2l ) Nmentioning

confidence: 99%

“…Unlike Observation 6, we cannot directly apply the analogous result of Gruslys [5]. Instead, we adapt his method to our setting.…”

Section: Lower Bound On the Number Of Uncovered Verticesmentioning

confidence: 99%

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Partitioning the Boolean lattice into copies of a poset

Gruslys

Leader

Tomon

2019

Journal of Combinatorial Theory, Series A

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Partitioning the vertices of a torus into isomorphic subgraphs

Bonamy

Morrison

Scott

2020

Journal of Combinatorial Theory, Series A

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Let H be an induced subgraph of the torus C m k . We show that when k ≥ 3 is even and |V (H)| divides some power of k, then for sufficiently large n the torus C n k has a perfect vertex-packing with induced copies of H. On the other hand, disproving a conjecture of Gruslys, we show that when k is odd and not a prime power, then there exists H such that |V (H)| divides some power of k, but there is no n such that C n k has a perfect vertex-packing with copies of H. We also disprove a conjecture of Gruslys, Leader and Tan by exhibiting a subgraph H of the k-dimensional hypercube Q k , such that there is no n for which Q n has a perfect edge-packing with copies of H.

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Packing the Boolean lattice with copies of a poset

Tomon

2019

J. London Math. Soc.

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Let P be a partially ordered set. We prove that if n is sufficiently large, then there exists a packing P of copies of P in the Boolean lattice (2 [n] , ⊂) that covers almost every element of 2 [n] : P might not cover the minimum and maximum of 2 [n] , and at most |P | − 1 additional points due to divisibility. In particular, if |P | divides 2 n − 2, then the truncated Boolean lattice 2 [n] − {∅, [n]} can be partitioned into copies of P . 1 otherwise, there is no copy of P covering [n] or ∅ in 2 [n] 1 The second conjecture of Lonc is concerned with posets P which do not necessarily satisfy that the size of P is a power of 2, or have a unique minimum and maximum. In this case, it is still reasonable to believe that there exists a P -packing in 2 [n] that covers almost everything. More precisely, the conjecture states that if n is sufficiently large and 2 n − 2 is divisible by |P |, then the truncated Boolean lattice 2 [n] − {∅, [n]} has a P -partition. This conjecture was verified by Lonc [12] in the case P is an antichain, or |P | ≤ 4. Also, Gruslys, Leader and Tomon [7] proposed a relaxation of this conjecture. That is, there exists a constant c(P ) such that 2 [n] has a P -packing that covers all but at most c(P ) elements of 2 [n] for every n. This conjecture was verified by the author of this paper [16] in case P has a unique minimum and maximum (but size not necessarily a power of 2).We settle both of the aforementioned conjectures in the following theorem.Theorem 2. Let P be a poset. There exists n 0 = n 0 (P ) such that if n ≥ n 0 , then there exists a P -packing P of 2 [n] − {∅, [n]} such that the number of elements not covered by P is at most |P | − 1.Our paper is organized as follows. In the next subsections, we discuss some related partitioning results and we introduce our notation. In Section 2, we prove Theorem 1. In Section 3, we prove Theorem 2. We finish our paper with some remarks and open problems in Section 4.

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Decomposing the vertex set of a hypercube into isomorphic subgraphs

Cited by 5 publications

References 5 publications

Partitioning the Boolean lattice into copies of a poset

Partitioning the Boolean lattice into copies of a poset

Partitioning the vertices of a torus into isomorphic subgraphs

Packing the Boolean lattice with copies of a poset

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