74 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
Boxicity and Cubicity of Asteroidal Triple free graphs
An axis parallel -dimensional box is the Cartesian product where each is a closed interval on the real line.
The {\it boxicity} of a graph , denoted as \boxi(G), is the minimum
integer such that can be represented as the intersection graph of a
collection of -dimensional boxes. An axis parallel unit cube in
-dimensional space or a -cube is defined as the Cartesian product where each is a closed interval on the
real line of the form . The {\it cubicity} of , denoted as
\cub(G), is the minimum integer such that can be represented as the
intersection graph of a collection of -cubes.
Let denote a star graph on nodes. We define {\it claw number} of
a graph as the largest positive integer such that is an induced
subgraph of and denote it as \claw.
Let be an AT-free graph with chromatic number and claw number
\claw. In this paper we will show that \boxi(G) \leq \chi(G) and this bound
is tight. We also show that \cub(G) \leq \boxi(G)(\ceil{\log_2 \claw} +2)
\chi(G)(\ceil{\log_2 \claw} +2). If is an AT-free graph having
girth at least 5 then \boxi(G) \leq 2 and therefore \cub(G) \leq
2\ceil{\log_2 \claw} +4.Comment: 15 pages: We are replacing our earlier paper regarding boxicity of
permutation graphs with a superior result. Here we consider the boxicity of
AT-free graphs, which is a super class of permutation graph
Cubicity of interval graphs and the claw number
Let be a simple, undirected graph where is the set of vertices
and is the set of edges. A -dimensional cube is a Cartesian product
, where each is a closed interval of
unit length on the real line. The \emph{cubicity} of , denoted by \cub(G)
is the minimum positive integer such that the vertices in can be mapped
to axis parallel -dimensional cubes in such a way that two vertices are
adjacent in if and only if their assigned cubes intersect. Suppose
denotes a star graph on nodes. We define \emph{claw number} of
the graph to be the largest positive integer such that is an induced
subgraph of . It can be easily shown that the cubicity of any graph is at
least \ceil{\log_2\psi(G)}.
In this paper, we show that, for an interval graph
\ceil{\log_2\psi(G)}\le\cub(G)\le\ceil{\log_2\psi(G)}+2. Till now we are
unable to find any interval graph with \cub(G)>\ceil{\log_2\psi(G)}. We also
show that, for an interval graph , \cub(G)\le\ceil{\log_2\alpha}, where
is the independence number of . Therefore, in the special case of
, \cub(G) is exactly \ceil{\log_2\alpha}.
The concept of cubicity can be generalized by considering boxes instead of
cubes. A -dimensional box is a Cartesian product , where each is a closed interval on the real
line. The \emph{boxicity} of a graph, denoted , is the minimum
such that is the intersection graph of -dimensional boxes. It is clear
that box(G)\le\cub(G). From the above result, it follows that for any graph
, \cub(G)\le box(G)\ceil{\log_2\alpha}
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 of Line Graphs
Boxicity of a graph H, denoted by box(H), is the minimum integer k such that
H is an intersection graph of axis-parallel k-dimensional boxes in R^k. In this
paper, we show that for a line graph G of a multigraph, box(G) <=
2\Delta(\lceil log_2(log_2(\Delta)) \rceil + 3) + 1, where \Delta denotes the
maximum degree of G. Since \Delta <= 2(\chi - 1), for any line graph G with
chromatic number \chi, box(G) = O(\chi log_2(log_2(\chi))). For the
d-dimensional hypercube H_d, we prove that box(H_d) >= (\lceil log_2(log_2(d))
\rceil + 1)/2. The question of finding a non-trivial lower bound for box(H_d)
was left open by Chandran and Sivadasan in [L. Sunil Chandran and Naveen
Sivadasan. The cubicity of Hypercube Graphs. Discrete Mathematics,
308(23):5795-5800, 2008]. The above results are consequences of bounds that we
obtain for the boxicity of fully subdivided graphs (a graph which can be
obtained by subdividing every edge of a graph exactly once).Comment: 14 page
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