Cubicity, Boxicity and Vertex Cover
| dc.creator | Chandran, L. Sunil | |
| dc.creator | Das, Anita | |
| dc.creator | Shah, Chintan | |
| dc.date | 2007-12-17 | |
| dc.date.accessioned | 2026-07-07T08:49:39Z | |
| dc.date.available | 2026-07-07T08:49:39Z | |
| dc.description | A $k$-dimensional box is the cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $box(G)$, is the minimum integer $k$ such that $G$ is the intersection graph of a collection of $k$-dimensional boxes. A unit cube in $k$-dimensional space or a $k$-cube is defined as the cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line of the form $[a_i, a_{i}+1]$. The {\it cubicity} of $G$, denoted as $cub(G)$, is the minimum $k$ such that $G$ is the intersection graph of a collection of $k$-cubes. In this paper we show that $cub(G) \leq t + \left \lceil \log (n - t)\right\rceil - 1$ and $box(G) \leq \left \lfloor\frac{t}{2}\right\rfloor + 1$, where $t$ is the cardinality of the minimum vertex cover of $G$ and $n$ is the number of vertices of $G$. We also show the tightness of these upper bounds. F. S. Roberts in his pioneering paper on boxicity and cubicity had shown that for a graph $G$, $box(G) \leq \left \lfloor\frac{n}{2} \right \rfloor$, where $n$ is the number of vertices of $G$, and this bound is tight. We show that if $G$ is a bipartite graph then $box(G) \leq \left \lceil\frac{n}{4} \right\rceil$ and this bound is tight. We point out that there exist graphs of very high boxicity but with very low chromatic number. For example there exist bipartite (i.e., 2 colorable) graphs with boxicity equal to $\frac{n}{4}$. Interestingly, if boxicity is very close to $\frac{n}{2}$, then chromatic number also has to be very high. In particular, we show that if $box(G) = \frac{n}{2} - s$, $s \geq 0$, then $χ(G) \geq \frac{n}{2s+2}$, where $χ(G)$ is the chromatic number of $G$. | |
| dc.description | 12 pages | |
| dc.identifier | https://arxiv.org/abs/0712.2688 | |
| dc.identifier | http://arxiv.org/abs/0712.2688 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/144378 | |
| dc.subject | Combinatorics | |
| dc.title | Cubicity, Boxicity and Vertex Cover | |
| dc.type | text |