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 there exists an integer $k$ such that, for $n$ sufficiently large, 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 circulant graphs, rectangular and triangular grids.

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关于一些图类的平衡性

给定一个图 $G$，如果我们能找到 $G$ 的一个副本使得它的一半边在每个颜色类别中，则称 $K_n$ 的边的 2-着色包含 $G$ 的平衡副本. 如果存在一个整数 $k$，对于足够大的 $n$，每个颜色类别中具有超过 $k$ 边的 $K_n$ 的每个 2-着色包含 $G$ 的平衡副本，那么我们说$G$ 是可平衡的。平衡性是由 Caro、Hansberg 和 Montejano 引入的，他们还给出了可平衡图的结构特征。在本文中，我们通过寻找图是否可平衡的新充分条件来扩展可平衡性的研究。我们使用这些条件来充分表征图类的平衡性，例如循环图、矩形和三角形网格。