120 research outputs found
On (2,3)-agreeable Box Societies
The notion of -agreeable society was introduced by Deborah Berg et
al.: a family of convex subsets of is called -agreeable if any
subfamily of size contains at least one non-empty -fold intersection. In
that paper, the -agreeability of a convex family was shown to imply the
existence of a subfamily of size with non-empty intersection, where
is the size of the original family and is an explicit
constant depending only on and . The quantity is called
the minimal \emph{agreement proportion} for a -agreeable family in
.
If we only assume that the sets are convex, simple examples show that
for -agreeable families in where . In this paper,
we introduce new techniques to find positive lower bounds when restricting our
attention to families of -boxes, i.e. cuboids with sides parallel to the
coordinates hyperplanes. We derive explicit formulas for the first non-trivial
case: the case of -agreeable families of -boxes with .Comment: 15 pages, 10 figure
Boxicity and topological invariants
The boxicity of a graph is the smallest integer for which there
exist interval graphs , , such that . In the first part of this note, we prove that every graph on
edges has boxicity , which is asymptotically best
possible. We use this result to study the connection between the boxicity of
graphs and their Colin de Verdi\`ere invariant, which share many similarities.
Known results concerning the two parameters suggest that for any graph , the
boxicity of is at most the Colin de Verdi\`ere invariant of , denoted by
. We observe that every graph has boxicity , while there are graphs with boxicity . In the second part of this note, we focus on graphs embeddable on a
surface of Euler genus . We prove that these graphs have boxicity
, while some of these graphs have boxicity . This improves the previously best known upper and lower bounds.
These results directly imply a nearly optimal bound on the dimension of the
adjacency poset of graphs on surfaces.Comment: 6 page
Boxicity and Cubicity of Product Graphs
The 'boxicity' ('cubicity') of a graph G is the minimum natural number k such
that G can be represented as an intersection graph of axis-parallel rectangular
boxes (axis-parallel unit cubes) in . In this article, we give estimates
on the boxicity and the cubicity of Cartesian, strong and direct products of
graphs in terms of invariants of the component graphs. In particular, we study
the growth, as a function of , of the boxicity and the cubicity of the
-th power of a graph with respect to the three products. Among others, we
show a surprising result that the boxicity and the cubicity of the -th
Cartesian power of any given finite graph is in and
, respectively. On the other hand, we show that there
cannot exist any sublinear bound on the growth of the boxicity of powers of a
general graph with respect to strong and direct products.Comment: 14 page
Boxicity and separation dimension
A family of permutations of the vertices of a hypergraph is
called 'pairwise suitable' for if, for every pair of disjoint edges in ,
there exists a permutation in in which all the vertices in one
edge precede those in the other. The cardinality of a smallest such family of
permutations for is called the 'separation dimension' of and is denoted
by . Equivalently, is the smallest natural number so that
the vertices of can be embedded in such that any two
disjoint edges of can be separated by a hyperplane normal to one of the
axes. We show that the separation dimension of a hypergraph is equal to the
'boxicity' of the line graph of . This connection helps us in borrowing
results and techniques from the extensive literature on boxicity to study the
concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to
WG-2014. Some results proved in this paper are also present in
arXiv:1212.6756. arXiv admin note: substantial text overlap with
arXiv:1212.675
Local Boxicity, Local Dimension, and Maximum Degree
In this paper, we focus on two recently introduced parameters in the
literature, namely `local boxicity' (a parameter on graphs) and `local
dimension' (a parameter on partially ordered sets). We give an `almost linear'
upper bound for both the parameters in terms of the maximum degree of a graph
(for local dimension we consider the comparability graph of a poset). Further,
we give an time deterministic algorithm to compute a local box
representation of dimension at most for a claw-free graph, where
and denote the number of vertices and the maximum degree,
respectively, of the graph under consideration. We also prove two other upper
bounds for the local boxicity of a graph, one in terms of the number of
vertices and the other in terms of the number of edges. Finally, we show that
the local boxicity of a graph is upper bounded by its `product dimension'.Comment: 11 page
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