Cubicity of interval graphs and the claw number
| dc.creator | Adiga, Abhijin | |
| dc.creator | Chandran, L. Sunil | |
| dc.date | 2009-03-06 | |
| dc.date.accessioned | 2026-07-07T12:49:52Z | |
| dc.date.available | 2026-07-07T12:49:52Z | |
| dc.description | 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} $ψ(G)$ 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 $\ceil{\log_2ψ(G)}$. In this paper, we show that, for an interval graph $G$ $\ceil{\log_2ψ(G)}\le\cub(G)\le\ceil{\log_2ψ(G)}+2$. Till now we are unable to find any interval graph with $\cub(G)>\ceil{\log_2ψ(G)}$. We also show that, for an interval graph $G$, $\cub(G)\le\ceil{\log_2α}$, where $α$ is the independence number of $G$. Therefore, in the special case of $ψ(G)=α$, $\cub(G)$ is exactly $\ceil{\log_2α}$. 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 $G$, $\cub(G)\le box(G)\ceil{\log_2α}$. | |
| dc.identifier | https://arxiv.org/abs/0903.1197 | |
| dc.identifier | http://arxiv.org/abs/0903.1197 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/222519 | |
| dc.subject | Combinatorics | |
| dc.title | Cubicity of interval graphs and the claw number | |
| dc.type | text |