On the number of edges of restricted matchstick graphs

TL;DR

This paper investigates the maximum number of edges in planar, noncrossing unit-length graphs called matchstick graphs, providing bounds for triangle-free cases and graphs within a disk, revealing surprising limitations.

Contribution

It establishes new upper bounds on edges for matchstick graphs in two scenarios, advancing understanding of their structural limitations.

Findings

Triangle-free matchstick graphs have at most 2n - c√n edges.

Graphs within a disk have at most (2 - ε(r))n edges.

Bounds are tight up to certain constants.

Abstract

A graph whose vertices are points in the plane and whose edges are noncrossing straight-line segments of unit length is called a \emph{matchstick graph}. We prove two somewhat counterintuitive results concerning the maximum number of edges of such graphs in two different scenarios. First, we show that there is a constant $c>0$ such that every triangle-free matchstick graph on $n$ vertices has at most $2n-c\sqrt{n}$ edges. This statement is not true for any $c>\sqrt2.$ We also prove that for every $r>0$, there is a constant $\varepsilon(r)>0$ with the property that every matchstick graph on $n$ vertices contained in a disk of radius $r$ has at most $(2-\varepsilon(r))n$ edges.