Note on the number of edges in families with linear union-complexity

TL;DR

This paper establishes a bound on the number of edges in intersection graphs of planar set families with linear union-complexity, improving existing bounds for pseudo-discs and related geometric objects.

Contribution

It provides a simple argument to bound edges in such graphs and improves the chromatic number bound for pseudo-discs.

Findings

Number of edges in intersection graphs is O(ω(G)n) for families with linear union-complexity.

Chromatic number of intersection graphs of pseudo-discs is less than 19 times their clique number.

The results improve previous bounds on intersection graph properties for geometric set families.

Abstract

We give a simple argument showing that the number of edges in the intersection graph $G$ of a family of $n$ sets in the plane with a linear union-complexity is $O(\omega(G)n)$. In particular, we prove $\chi(G)\leq \text{col}(G)< 19\omega(G)$ for intersection graph $G$ of a family of pseudo-discs, which improves a previous bound.