36 research outputs found
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
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
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