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

Boxicity and topological invariants

The boxicity of a graph $G=(V,E)$ is the smallest integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap \cdots \cap E_k$. In the first part of this note, we prove that every graph on $m$ edges has boxicity $O(\sqrt{m \log m})$, 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 $G$, the boxicity of $G$ is at most the Colin de Verdi\`ere invariant of $G$, denoted by $\mu(G)$. We observe that every graph $G$ has boxicity $O(\mu(G)^4(\log \mu(G))^2)$, while there are graphs $G$ with boxicity $\Omega(\mu(G)\sqrt{\log \mu(G)})$. In the second part of this note, we focus on graphs embeddable on a surface of Euler genus $g$. We prove that these graphs have boxicity $O(\sqrt{g}\log g)$, while some of these graphs have boxicity $\Omega(\sqrt{g \log g})$. 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

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 $O(n\Delta^2)$ time deterministic algorithm to compute a local box representation of dimension at most $3\Delta$ for a claw-free graph, where $n$ and $\Delta$ 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 of graphs on surfaces

The boxicity of a graph $G=(V,E)$ is the least integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap ... \cap E_k$. Scheinerman proved in 1984 that outerplanar graphs have boxicity at most two and Thomassen proved in 1986 that planar graphs have boxicity at most three. In this note we prove that the boxicity of toroidal graphs is at most 7, and that the boxicity of graphs embeddable in a surface $\Sigma$ of genus $g$ is at most $5g+3$. This result yields improved bounds on the dimension of the adjacency poset of graphs on surfaces.Comment: 9 pages, 2 figure

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