On difference graphs and the local dimension of posets

Kim, Jinha; Martin, Ryan; Masařík, Tomáš; Shull, Warren; Smith, Heather; Uzzell, Andrew J.; Wang, Zhiyu

doi:10.1016/j.ejc.2019.103074

Cited by 14 publications

(30 citation statements)

References 17 publications

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“…Can any of the bounds in Proposition 2 be improved? Kim et al [7] asked whether or not ldim Q n = n for all n. We believe that this is not the case, and that our best lower bound is asymptotically correct. We may even propose the following stronger conjecture.…”

Section: Problemsmentioning

confidence: 85%

“…Wang Zhiyu [18], following a comment by Christophe Crespelle in [7], suggested that an information entropy method might help improve the bounds on ldim Q n .…”

Section: Lower Boundsmentioning

confidence: 99%

“…Conversely, there exists a prefix-free code C such that the expected length of C(X) is at most H(X) log |Ω| + 1. In a note added to the end of [7], Crespelle suggested a method of encoding an arbitrary poset on a fixed ground set, which we call the Crespelle code. Given a ground set X with cardinality n and a poset P on X, we encode P as a word over a 3n-symbol alphabet Ω as follows.…”

Section: Lower Boundsmentioning

confidence: 99%

“…Kim, Martin, Masařík, Shull, Smith, Uzzell, and Wang [7] proved that the maximum local dimension of a poset on n points is at between…”

Section: Lower Boundsmentioning

confidence: 99%

“…Kim, Martin, Masařík, Shull, Smith, Uzzell, and Wang [7] proved that ldim Q n is at least Ω n log n , but so far the only upper bound we have for ldim Q n is the trivial bound of n.…”

Section: Introductionmentioning

confidence: 96%

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The local dimension of suborders of the Boolean lattice

Lewis

2020

Preprint

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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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Section: Problemsmentioning

confidence: 85%

“…Wang Zhiyu [18], following a comment by Christophe Crespelle in [7], suggested that an information entropy method might help improve the bounds on ldim Q n .…”

Section: Lower Boundsmentioning

confidence: 99%

Section: Lower Boundsmentioning

confidence: 99%

“…Kim, Martin, Masařík, Shull, Smith, Uzzell, and Wang [7] proved that the maximum local dimension of a poset on n points is at between…”

Section: Lower Boundsmentioning

confidence: 99%

“…Kim, Martin, Masařík, Shull, Smith, Uzzell, and Wang [7] proved that ldim Q n is at least Ω n log n , but so far the only upper bound we have for ldim Q n is the trivial bound of n.…”

Section: Introductionmentioning

confidence: 96%

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The local dimension of suborders of the Boolean lattice

Lewis

2020

Preprint

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Local and Union Page Numbers

Merker

Ueckerdt

2019

Lecture Notes in Computer Science

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We introduce the novel concepts of local and union book embeddings, and, as the corresponding graph parameters, the local page number pn (G) and the union page number pn u (G). Both parameters are relaxations of the classical page number pn(G), and for every graph G we have pn (G) pn u (G) pn(G). While for pn(G) one minimizes the total number of pages in a book embedding of G, for pn (G) we instead minimize the number of pages incident to any one vertex, and for pn u (G) we instead minimize the size of a partition of G with each part being a vertex-disjoint union of crossing-free subgraphs. While pn (G) and pn u (G) are always within a multiplicative factor of 4, there is no bound on the classical page number pn(G) in terms of pn (G) or pn u (G). We show that local and union page numbers are closer related to the graph's density, while for the classical page number the graph's global structure can play a much more decisive role. We introduce tools to investigate local and union book embeddings in exemplary considerations of the class of all planar graphs and the class of graphs of tree-width k.As an incentive to pursue research in this new direction, we offer a list of intriguing open problems.

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