Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count

TL;DR

This paper investigates properties of triangle-free penny graphs, establishing bounds on their degeneracy, choosability, diameter, and edge count, revealing structural limitations and characteristics of these geometric graphs.

Contribution

The paper provides new bounds on degeneracy, choosability, diameter, and edges for triangle-free penny graphs, advancing understanding of their structural properties.

Findings

Degeneracy of at most two

List coloring number (choosability) at most three

Diameter at least proportional to ( n)

Abstract

We show that triangle-free penny graphs have degeneracy at most two, list coloring number (choosability) at most three, diameter $D=\Omega(\sqrt n)$, and at most $\min\bigl(2n-\Omega(\sqrt n),2n-D-2\bigr)$ edges.