On the balanceability of some graph classes

TL;DR

This paper investigates the concept of balanceability in graphs, providing new conditions and fully characterizing the balanceability of certain graph classes like grids and circulant graphs.

Contribution

It extends the study of balanceability by establishing new sufficient conditions and offers a complete characterization for specific graph classes.

Findings

Characterized the balanceability of rectangular and triangular grids.

Identified new sufficient conditions for a graph to be balanceable.

Provided a full classification of balanceability for a class of circulant graphs.

Abstract

Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges are in each color class. If, for every sufficiently large $n$, there exists an integer $k$ such that every 2-coloring of $K_n$ with more than $k$ edges in each color class contains a balanced copy of $G$, then we say that $G$ is balanceable. Balanceability was introduced by Caro, Hansberg and Montejano, who also gave a structural characterization of balanceable graphs. In this paper, we extend the study of balanceability by finding new sufficient conditions for a graph to be balanceable or not. We use those conditions to fully characterize the balanceability of graph classes such as rectangular and triangular grids, as well as a special class of circulant graphs.