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Boxicity and Maximum degree

L. Sunil Chandran·Mathew C. Francis·Naveen Sivadasan
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Abstract

An axis-parallel $d$--dimensional box is a Cartesian product $R_1 \times R_2 \times ... \times R_d$ where $R_i$ (for $1 \le i \le d$) is a closed interval of the form $[a_i, b_i]$ on the real line. For a graph $G$, its \emph{boxicity} $\boxi(G)$ is the minimum dimension $d$, such that $G$ is representable as the intersection graph of (axis--parallel) boxes in $d$--dimensional space. The concept of boxicity finds applications in various areas such as ecology, operation research etc. We show that for any graph $G$ with maximum degree $Δ$, $\boxi(G) \le 2 Δ^2 + 2$. That the bound does not depend on the number of vertices is a bit surprising considering the fact that there are highly connected bounded degree graphs such as expander graphs. Our proof is very short and constructive. We conjecture that $\boxi(G)$ is $O(Δ)$.

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L. Sunil Chandran, Mathew C. Francis, Naveen Sivadasan. 2006-10-08. Boxicity and Maximum degree. https://arxiv.org/abs/math/0610262

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