Balanced Cayley graphs and balanced planar graphs

TL;DR

This paper classifies balanced Cayley graphs on abelian groups, proves the non-existence of cubic balanced planar graphs, and discusses conjectures related to balanced regular graphs, connecting graph theory with algebraic and geometric properties.

Contribution

It provides a complete classification of balanced Cayley graphs on abelian groups and establishes the non-existence of cubic balanced planar graphs.

Findings

Complete classification of balanced Cayley graphs on abelian groups

Proof that no cubic balanced planar graphs exist

Presentation of conjectures on balanced regular graphs

Abstract

A balanced graph is a bipartite graph with no induced circuit of length 2 mod 4. These graphs arise in linear programming. We focus on graph-algebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in Section 5 of this paper, we prove that there is no cubic balanced planar graph. Finally, some remarkable conjectures for balanced regular graphs are also presented.