# Small 4-regular planar graphs that are not circle representable

## TL;DR

This paper presents smaller counterexamples to Lovász's conjecture that all 4-regular planar graphs are circle representable, reducing the smallest known counterexample size significantly.

## Contribution

It introduces the smallest known simple 4-regular planar graphs that are not circle representable, decreasing the previous minimal order from 822 to 68.

## Key findings

- Counterexamples of size 68 exist among simple graphs.
- A multigraph counterexample of size 12 was used to find smaller simple graphs.
- Lovász's conjecture is disproved by these smaller counterexamples.

## Abstract

A 4-regular planar graph $G$ is said to be circle representable if there exists a collection of circles drawn on the plane such that the touching and crossing points correspond to the vertices of $G$, and the circular arcs between those points correspond to the edges of $G$. Lov\'asz (1970) conjectured that every 4-regular planar graph has a circle representation, but an infinite family of counterexamples was given by Bekos and Raftopoulou (2015). We reduce the order of the smallest known counterexamples among simple graphs from 822 to 68 based on a multigraph counterexample of order 12.