Boxicity, poset dimension, and excluded minors

TL;DR

This paper establishes improved bounds on representing graphs with no $K_t$-minor using interval and circular-arc graphs, linking boxicity and poset dimension to coloring parameters.

Contribution

It provides tighter bounds on the intersection representations of graphs excluding $K_t$-minors, connecting boxicity, poset dimension, and coloring parameters.

Findings

Graphs with no $K_t$-minor can be represented as the intersection of $O(t^2 \log t)$ interval graphs.

Such graphs can also be represented as the intersection of $rac{15}{2} t^2$ circular-arc graphs.

Improves previous bounds from $O(t^4)$ to $O(t^2 \log t)$.

Abstract

In this short note, we relate the boxicity of graphs (and the dimension of posets) with their generalized coloring parameters. In particular, together with known estimates, our results imply that any graph with no $K_t$-minor can be represented as the intersection of $O(t^2\log t)$ interval graphs (improving the previous bound of $O(t^4)$), and as the intersection of $\tfrac{15}2 t^2$ circular-arc graphs.