2,883 research outputs found
Minors and dimension
It has been known for 30 years that posets with bounded height and with cover
graphs of bounded maximum degree have bounded dimension. Recently, Streib and
Trotter proved that dimension is bounded for posets with bounded height and
planar cover graphs, and Joret et al. proved that dimension is bounded for
posets with bounded height and with cover graphs of bounded tree-width. In this
paper, it is proved that posets of bounded height whose cover graphs exclude a
fixed topological minor have bounded dimension. This generalizes all the
aforementioned results and verifies a conjecture of Joret et al. The proof
relies on the Robertson-Seymour and Grohe-Marx graph structure theorems.Comment: Updated reference
On the Duality of Semiantichains and Unichain Coverings
We study a min-max relation conjectured by Saks and West: For any two posets
and the size of a maximum semiantichain and the size of a minimum
unichain covering in the product are equal. For positive we state
conditions on and that imply the min-max relation. Based on these
conditions we identify some new families of posets where the conjecture holds
and get easy proofs for several instances where the conjecture had been
verified before. However, we also have examples showing that in general the
min-max relation is false, i.e., we disprove the Saks-West conjecture.Comment: 10 pages, 3 figure
Hereditary properties of combinatorial structures: posets and oriented graphs
A hereditary property of combinatorial structures is a collection of
structures (e.g. graphs, posets) which is closed under isomorphism, closed
under taking induced substructures (e.g. induced subgraphs), and contains
arbitrarily large structures. Given a property P, we write P_n for the
collection of distinct (i.e., non-isomorphic) structures in a property P with n
vertices, and call the function n -> |P_n| the speed (or unlabelled speed) of
P. Also, we write P^n for the collection of distinct labelled structures in P
with vertices labelled 1,...,n, and call the function n -> |P^n| the labelled
speed of P.
The possible labelled speeds of a hereditary property of graphs have been
extensively studied, and the aim of this paper is to investigate the possible
speeds of other combinatorial structures, namely posets and oriented graphs.
More precisely, we show that (for sufficiently large n), the labelled speed of
a hereditary property of posets is either 1, or exactly a polynomial, or at
least 2^n - 1. We also show that there is an initial jump in the possible
unlabelled speeds of hereditary properties of posets, tournaments and directed
graphs, from bounded to linear speed, and give a sharp lower bound on the
possible linear speeds in each case.Comment: 26 pgs, no figure
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