Upper bound on cubicity in terms of boxicity for graphs of low chromatic number

The boxicity (respectively cubicity) of a graph $G$ is the minimum non-negative integer $k$, such that $G$ can be represented as an intersection graph of axis-parallel $k$-dimensional boxes (respectively $k$-dimensional unit cubes) and is denoted by $box(G)$ (respectively $cub(G)$). It was shown by Adiga and Chandran (Journal of Graph Theory, 65(4), 2010) that for any graph $G$, $cub(G) \le$ box$(G) \left \lceil \log_2 \alpha \right \rceil$, where $\alpha = \alpha(G)$ is the cardinality of the maximum independent set in $G$. In this note we show that $cub(G) \le 2 \left \lceil \log_2 \chi(G) \right \rceil box(G) + \chi(G) \left \lceil \log_2 \alpha(G) \right \rceil$. In general, this result can provide a much better upper bound than that of Adiga and Chandran for graph classes with bounded chromatic number. For example, for bipartite graphs we get, $cub(G) \le 2 (box(G) + \left \lceil \log_2 \alpha(G) \right \rceil )$. Moreover we show that for every positive integer $k$, there exist graphs with chromatic number $k$, such that for every $\epsilon > 0$, the value given by our upper bound is at most $(1+\epsilon)$ times their cubicity. Thus, our upper bound is almost tight

A sufficient condition for a graph with boxicity at most its chromatic number

A box in Euclidean $k$-space is the Cartesian product of $k$ closed intervals on the real line. The boxicity of a graph $G$, denoted by $\text{box}(G)$, is the minimum nonnegative integer $k$ such that $G$ can be isomorphic to the intersection graph of a family of boxes in Euclidean $k$-space. In this paper, we present a sufficient condition for a graph $G$ under which $\text{box}(G)\leq \chi (G)$ holds, where $\chi (G)$ denotes the chromatic number of $G$. Bhowmick and Chandran (2010) proved that $\text{box}(G)\leq \chi (G)$ holds for a graph $G$ with no asteroidal triples. We prove that $\text{box}(G)\leq \chi (G)$ holds for a graph $G$ in a special family of circulant graphs with an asteroidal triple.Comment: 10 pages, 4 figure

On the stab number of rectangle intersection graphs

We introduce the notion of \emph{stab number} and \emph{exact stab number} of rectangle intersection graphs, otherwise known as graphs of boxicity at most 2. A graph $G$ is said to be a \emph{$k$-stabbable rectangle intersection graph}, or \emph{$k$-SRIG} for short, if it has a rectangle intersection representation in which $k$ horizontal lines can be chosen such that each rectangle is intersected by at least one of them. If there exists such a representation with the additional property that each rectangle intersects exactly one of the $k$ horizontal lines, then the graph $G$ is said to be a \emph{$k$-exactly stabbable rectangle intersection graph}, or \emph{$k$-ESRIG} for short. The stab number of a graph $G$, denoted by $stab(G)$, is the minimum integer $k$ such that $G$ is a $k$-SRIG. Similarly, the exact stab number of a graph $G$, denoted by $estab(G)$, is the minimum integer $k$ such that $G$ is a $k$-ESRIG. In this work, we study the stab number and exact stab number of some subclasses of rectangle intersection graphs. A lower bound on the stab number of rectangle intersection graphs in terms of its pathwidth and clique number is shown. Tight upper bounds on the exact stab number of split graphs with boxicity at most 2 and block graphs are also given. We show that for $k\leq 3$, $k$-SRIG is equivalent to $k$-ESRIG and for any $k\geq 10$, there is a tree which is a $k$-SRIG but not a $k$-ESRIG. We also develop a forbidden structure characterization for block graphs that are 2-ESRIG and trees that are 3-ESRIG, which lead to polynomial-time recognition algorithms for these two classes of graphs. These forbidden structures are natural generalizations of asteroidal triples. Finally, we construct examples to show that these forbidden structures are not sufficient to characterize block graphs that are 3-SRIG or trees that are $k$-SRIG for any $k\geq 4$