Bounds for the boxicity of Mycielski graphs

TL;DR

This paper investigates bounds on the boxicity of Mycielski graphs, providing new inequalities and conditions that relate the boxicity of a graph to its Mycielski extension, with implications for understanding graph intersection representations.

Contribution

The paper establishes bounds for the boxicity of Mycielski graphs based on universal vertices and the edge clique cover number, and explores specific cases where these bounds are tight.

Findings

Bounds for boxicity of Mycielski graphs are derived.

Boxicity increases by at least half the number of universal vertices.

Relations between different Mycielski graph variants are analyzed.

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 $M_3(G)$.