On Covering Numbers, Young Diagrams, and the Local Dimension of Posets

Damásdi, Gábor; Felsner, Stefan; Girão, António; Keszegh, Balázs; Lewis, David; Nagy, Dániel T.; Ueckerdt, Torsten

doi:10.48550/arxiv.2001.06367

Cited by 3 publications

(4 citation statements)

References 12 publications

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“…The following lower bound for the local dimension of Q n 1,2 was proved by Girão and Lewis, and the proof was included in [3].…”

Section: Posets Whose Dimension and Local Dimension Are Equalmentioning

confidence: 98%

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The local dimension of suborders of the Boolean lattice

Lewis

2020

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We prove upper and lower bounds on the local dimension of any pair of layers of the Boolean lattice, and show that ldim Q n 1,⌊n/2⌋ ∼ n log 2 n as n → ∞. Previously, all that was known was a lower bound of Ω(n/ log n) and an upper bound of n.Improving a result of Kim, Martin, Masařík, Shull, Smith, Uzzell, and Wang, we also prove that that the maximum local dimension of an n-element poset is at least. We also show that there exist posets of arbitrarily large dimension whose dimension and local dimension are equal.

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“…The following lower bound for the local dimension of Q n 1,2 was proved by Girão and Lewis, and the proof was included in [3].…”

Section: Posets Whose Dimension and Local Dimension Are Equalmentioning

confidence: 98%

“…The bound ldim Q n ≥ (1 + o(1)) n log n , which improves one of Kim et al's results by a constant factor, was also proved by Stefan Felsner by different means. Felsner's proof can be found in [3].…”

Section: Lower Boundsmentioning

confidence: 99%

The local dimension of suborders of the Boolean lattice

Lewis

2020

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“…Damásdi, Felsner, Girão, Keszegh, Lewis, Nagy, and Ueckerdt [6] proved that a difference graph with n steps has local complete bipartite covering number equal to…”

Section: Local 2-dimension and Complete Bipartite Edge-coverings Of G...mentioning

confidence: 99%

Local $t$-dimension

Lewis

2020

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“…Using a connection between the (local) dimension of partially ordered sets and the (local) boxicity of graphs [1,27], Theorem 1.1 directly implies that partially ordered sets whose comparability graphs have maximum degree ∆ have local dimension O(∆), and this bound is optimal up to a constant factor (see [9] for more details and results on the local dimension of posets).…”

Section: Introductionmentioning

confidence: 99%

Local boxicity

Esperet,

Lichev

2020

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A box is the cartesian product of real intervals, which are either bounded or equal to R. A box is said to be d-local if at most d of the intervals are bounded. In this paper, we investigate the recently introduced local boxicity of a graph G, which is the minimum d such that G can be represented as the intersection of d-local boxes in some dimension. We prove that all graphs of maximum degree ∆ have local boxicity O(∆), while almost all graphs of maximum degree ∆ have local boxicity Ω(∆), improving known upper and lower bounds. We also give improved bounds on the local boxicity as a function of the number of edges or the genus. Finally, we investigate local boxicity through the lens of chromatic graph theory. We prove that the family of graphs of local boxicity at most 2 is χ-bounded, which means that the chromatic number of the graphs in this class can be bounded by a function of their clique number. This extends a classical result on graphs of boxicity at most 2.

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On Covering Numbers, Young Diagrams, and the Local Dimension of Posets

Cited by 3 publications

References 12 publications

The local dimension of suborders of the Boolean lattice

The local dimension of suborders of the Boolean lattice

Local $t$-dimension

Local boxicity

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