Bounds for the boxicity of Mycielski graphs

Abstract: A box in Euclidean $k$-space is the Cartesian product $I_1\times I_2\times \cdots \times I_k$, where $I_j$ is a closed interval 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. Mycielski introduced an interesting graph operation that extends a graph $G$ to a new graph $M(G)$, called the Mycielski graph of $G$. In this paper, we observe behavior of the boxicity of Mycielski graphs. The inequality $\text{box}(M(G))\geq \text{box}(G)$ holds for a graph $G$, and hence we are interested in whether the boxicity of the Mycielski graph of $G$ is more than that of $G$ or not. Here we give bounds for the boxicity of Mycielski graphs: for a graph $G$ with $l$ universal vertices, the inequalities $\text{box}(G)+\left \lceil \frac{l}{2}\right \rceil \leq \text{box}(M(G))\leq \theta (\overline{G})+\left\lceil \frac{l}{2}\right\rceil +1$ hold, where $\theta (\overline{G})$ is the edge clique cover number of the complement $\overline{G}$. Further observations determine the boxicity of the Mycielski graph $M(G)$, if $G$ has no universal vertices or odd universal vertices and satisfies $\text{box}(G)=\theta (\overline{G})$. We also present relations between the Mycielski graph $M(G)$ and its analogous ones $M_3(G)$ and $M_r(G)$ in the context of boxicity, which will encourage us to calculate the boxicity of $M(G)$ or $M_3(G)$.