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

Cubicity of interval graphs and the claw number

Let $G(V,E)$ be a simple, undirected graph where $V$ is the set of vertices and $E$ is the set of edges. A $b$-dimensional cube is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval of unit length on the real line. The \emph{cubicity} of $G$, denoted by \cub(G) is the minimum positive integer $b$ such that the vertices in $G$ can be mapped to axis parallel $b$-dimensional cubes in such a way that two vertices are adjacent in $G$ if and only if their assigned cubes intersect. Suppose $S(m)$ denotes a star graph on $m+1$ nodes. We define \emph{claw number} $\psi(G)$ of the graph to be the largest positive integer $m$ such that $S(m)$ is an induced subgraph of $G$. 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 $G$ \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 $G$, \cub(G)\le\ceil{\log_2\alpha}, where $\alpha$ is the independence number of $G$. Therefore, in the special case of $\psi(G)=\alpha$, \cub(G) is exactly \ceil{\log_2\alpha}. The concept of cubicity can be generalized by considering boxes instead of cubes. A $b$-dimensional box is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval on the real line. The \emph{boxicity} of a graph, denoted $box(G)$, is the minimum $k$ such that $G$ is the intersection graph of $k$-dimensional boxes. It is clear that box(G)\le\cub(G). From the above result, it follows that for any graph $G$, \cub(G)\le box(G)\ceil{\log_2\alpha}

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 $R^k$. 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 $d$, of the boxicity and the cubicity of the $d$-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 $d$-th Cartesian power of any given finite graph is in $O(\log d / \log\log d)$ and $\theta(d / \log d)$, 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

An upper bound for Cubicity in terms of Boxicity

AbstractAn axis-parallel b-dimensional box is a Cartesian product R1×R2×⋯×Rb where each Ri (for 1≤i≤b) is a closed interval of the form [ai,bi] on the real line. The boxicity of any graph G, box(G) is the minimum positive integer b such that G can be represented as the intersection graph of axis-parallel b-dimensional boxes. A b-dimensional cube is a Cartesian product R1×R2×⋯×Rb, where each Ri (for 1≤i≤b) is a closed interval of the form [ai,ai+1] on the real line. When the boxes are restricted to be axis-parallel cubes in b-dimension, the minimum dimension b required to represent the graph is called the cubicity of the graph (denoted by cub(G)). In this paper we prove that cub(G)≤⌈log2n⌉box(G), where n is the number of vertices in the graph. We also show that this upper bound is tight.Some immediate consequences of the above result are listed below: 1.Planar graphs have cubicity at most 3⌈log2n⌉.2.Outer planar graphs have cubicity at most 2⌈log2n⌉.3.Any graph of treewidth tw has cubicity at most (tw+2)⌈log2n⌉. Thus, chordal graphs have cubicity at most (ω+1)⌈log2n⌉ and circular arc graphs have cubicity at most (2ω+1)⌈log2n⌉, where ω is the clique number.The above upper bounds are tight, but for small constant factors

The cubicity of hypercube graphs

For a graph G, its cubicity View the MathML source is the minimum dimension k such that Gis representable as the intersection graph of (axis-parallel) cubes in k-dimensional space. (A k-dimensional cube is a Cartesian product R1×R2×cdots, three dots, centered×Rk, where Ri is a closed interval of the form [ai,ai+1] on the real line.) Chandran et al. [L.S. Chandran, C. Mannino, G. Oriolo, On the cubicity of certain graphs, Information Processing Letters 94 (2005) 113–118] showed that for a d-dimensional hypercube Hd, View the MathML source. In this paper, we use the probabilistic method to show that View the MathML source. The parameter boxicity generalizes cubicity: the boxicity View the MathML source of a graph G is defined as the minimum dimension k such that G is representable as the intersection graph of axis-parallel boxes in k-dimensional space. Since View the MathML source for any graph G, our result implies that View the MathML source. The problem of determining a non-trivial lower bound for View the MathML source is left open

The cubicity of hypercube graphs

The cubicity of hypercube graph

Boxicity of Line Graphs

The boxicity of a graph H, denoted by View the MathML source, is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in View the MathML source. In this paper we show that for a line graph G of a multigraph, View the MathML source, where Δ(G) denotes the maximum degree of G. Since G is a line graph, Δ(G)≤2(χ(G)−1), where χ(G) denotes the chromatic number of G, and therefore, View the MathML source. For the d-dimensional hypercube Qd, we prove that View the MathML source. The question of finding a nontrivial lower bound for View the MathML source was left open by Chandran and Sivadasan in [L. Sunil Chandran, Naveen Sivadasan, The cubicity of Hypercube Graphs. Discrete Mathematics 308 (23) (2008) 5795–5800]. The above results are consequences of bounds that we obtain for the boxicity of a fully subdivided graph (a graph that can be obtained by subdividing every edge of a graph exactly once)

Boxicity of line graphs

The 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 (G)(inverted right perpendicularlog(2) log(2) Delta(G)inverted left perpendicular + 3) + 1, where Delta(G) denotes the maximum degree of G. Since G is a line graph, Delta(G) <= 2(chi (G) - 1), where chi (G) denotes the chromatic number of G, and therefore, box(G) = 0(chi (G) log(2) log(2) (chi (G))). For the d-dimensional hypercube Q(d), we prove that box(Q(d)) >= 1/2 (inverted right perpendicularlog(2) log(2) dinverted left perpendicular + 1). The question of finding a nontrivial lower bound for box(Q(d)) was left open by Chandran and Sivadasan in [L. Sunil Chandran, Naveen Sivadasan, The cubicity of Hypercube Graphs. Discrete Mathematics 308 (23) (2008) 5795-5800]. The above results are consequences of bounds that we obtain for the boxicity of a fully subdivided graph (a graph that can be obtained by subdividing every edge of a graph exactly once). (C) 2011 Elsevier B.V. All rights reserved