On a Conjecture of Lov\'asz on Circle-Representations of Simple 4-Regular Planar Graphs

TL;DR

This paper proves Lovász's conjecture that every 3-connected simple 4-regular planar graph can be represented by a system of circles, providing bounds on the number of circles needed and showing some non-3-connected graphs cannot be realized this way.

Contribution

The paper confirms Lovász's conjecture for 3-connected graphs, establishes bounds on circle realizations, and identifies classes of graphs that cannot be realized as circle systems.

Findings

Confirmed Lovász's conjecture for 3-connected graphs

Provided bounds on the number of circles needed

Identified non-3-connected graphs that cannot be realized as circle systems

Abstract

Lov\'asz conjectured that every connected 4-regular planar graph G admits a realization as a system of circles, i.e., it can be drawn on the plane utilizing a set of circles, such that the vertices of G correspond to the intersection and touching points of the circles and the edges of G are the arc segments among pairs of intersection and touching points of the circles. In this paper, we settle this conjecture. In particular, (a) we first provide tight upper and lower bounds on the number of circles needed in a realization of any simple 4-regular planar graph, (b) we affirmatively answer Lov\'asz's conjecture, if G is 3-connected, and (c) we demonstrate an infinite class of simple connected 4-regular planar graphs which are not 3-connected (i.e., either simply connected or biconnected) and do not admit realizations as a system of circles.