On the balanceability of some graph classes
Abstract
Given a graph G, a 2-coloring of the edges of Kn 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 Kn 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.