178 research outputs found
Boolean Dimension, Components and Blocks
We investigate the behavior of Boolean dimension with respect to components
and blocks. To put our results in context, we note that for Dushnik-Miller
dimension, we have that if for every component of a poset
, then ; also if for every block
of a poset , then . By way of constrast, local dimension is
well behaved with respect to components, but not for blocks: if
for every component of a poset , then
; however, for every , there exists a poset
with and for every block of . In this
paper we show that Boolean dimension behaves like Dushnik-Miller dimension with
respect to both components and blocks: if for every
component of , then ; also if
for every block of , then .Comment: 12 pages. arXiv admin note: text overlap with arXiv:1712.0609
Dimension and cut vertices: an application of Ramsey theory
Motivated by quite recent research involving the relationship between the
dimension of a poset and graph-theoretic properties of its cover graph, we show
that for every , if is a poset and the dimension of a subposet
of is at most whenever the cover graph of is a block of the cover
graph of , then the dimension of is at most . We also construct
examples which show that this inequality is best possible. We consider the
proof of the upper bound to be fairly elegant and relatively compact. However,
we know of no simple proof for the lower bound, and our argument requires a
powerful tool known as the Product Ramsey Theorem. As a consequence, our
constructions involve posets of enormous size.Comment: Final published version with updated reference
A characterization of robert's inequality for boxicity
AbstractF.S. Roberts defined the boxicity of a graph G as the smallest positive integer n for which there exists a function F assigning to each vertex x ϵG a sequence F(x)(1),F(x)(2),…, F(x)(n) of closed intervals of R so that distinct vertices x and y are adjacent in G if and only if F(x)(i)∩F(y)(i)≠∅ for i = 1, 2, 3, …, n. Roberts then proved that if G is a graph having 2n + 1 vertices, then the boxicity of G is at most n. In this paper, we provide an explicit characterization of this inequality by determining for each n ⩾ 1 the minimum collection Cn of graphs so that a graph G having 2n + 1 vertices has boxicity n if and only if it contains a graph from Cn as an induced subgraph. We also discuss combinatorial connections with analogous characterization problems for rectangle graphs, circular arc graphs, and partially ordered sets
Embedding finite posets in cubes
AbstractIn this paper we define the n-cube Qn as the poset obtained by taking the cartesian product of n chains each consisting of two points. For a finite poset X, we then define dim2 X as the smallest positive integer n such that X can be embedded as a subposet of Qn. For any poset X we then have log2 |X| ⩽ dim2 X ⩽ |X|. For the distributive lattice L = 2 X, dim2 L = |X| and for the crown Skn, dim2 (Skn) = n + k. For each k ⩾ 2, there exist positive constants c1 and c2 so that for the poset X consisting of all one element and k-element subsets of an n-element set, the inequality c1 log2 n < dim2(X) < c2 log2 n holds for all n with k < n. A poset is called Q-critical if dim2 (X − x) < dim2(X) for every x ϵ X. We define a join operation ⊕ on posets under which the collection Q of all Q-critical posets which are not chains forms a semigroup in which unique factorization holds. We then completely determine the subcollection M ⊆ Q consisting of all posets X for which dim2 (X) = |X|
Burling graphs, chromatic number, and orthogonal tree-decompositions
A classic result of Asplund and Gr\"unbaum states that intersection graphs of
axis-aligned rectangles in the plane are -bounded. This theorem can be
equivalently stated in terms of path-decompositions as follows: There exists a
function such that every graph that has two
path-decompositions such that each bag of the first decomposition intersects
each bag of the second in at most vertices has chromatic number at most
. Recently, Dujmovi\'c, Joret, Morin, Norin, and Wood asked whether this
remains true more generally for two tree-decompositions. In this note we
provide a negative answer: There are graphs with arbitrarily large chromatic
number for which one can find two tree-decompositions such that each bag of the
first decomposition intersects each bag of the second in at most two vertices.
Furthermore, this remains true even if one of the two decompositions is
restricted to be a path-decomposition. This is shown using a construction of
triangle-free graphs with unbounded chromatic number due to Burling, which we
believe should be more widely known.Comment: v3: minor changes made following comments by the referees, v2: minor
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