We continue the study of balanceable graphs, defined by Caro, Hansberg, and Montejano in 2021 as graphs $G$ such that any $2$-coloring of the edges of a sufficiently large complete graph containing sufficiently many edges of each color contains a balanced copy of $G$. While the problem of recognizing balanceable graphs was conjectured to be NP-complete by Dailly, Hansberg, and Ventura in 2021, balanceable graphs admit an elegant combinatorial characterization: a graph is balanceable if and only there exist two vertex subsets, one containing half of all the graph's edges and another one such that the corresponding cut contains half of all the graph's edges. We consider a special case of this property, namely when one of the two sets is a vertex cover, and call the corresponding graphs simply balanceable. We prove a number of results on balanceable and simply balanceable regular graphs. First, we characterize simply balanceable regular graphs via a condition involving the independence number of the graph. Second, we address a question of Dailly, Hansberg, and Ventura from 2021 and show that every cubic graph is balanceable. Third, using Brooks' theorem, we show that every $4$-regular graph with order divisible by $4$ is balanceable. Finally, we show that it is NP-complete to determine if a $9$-regular graph is simply balanceable.

翻译：我们继续研究由Caro、Hansberg和Montejano于2021年定义的可平衡图：图G满足性质，在充分大的完全图中对边进行2-染色，且每种颜色出现足够多的边时，必然包含一个与G同构的平衡子图。尽管Dailly、Hansberg和Ventura在2021年推测识别可平衡图的问题是NP完全的，但可平衡图具有优雅的组合刻画：一个图是可平衡的当且仅当存在两个顶点子集，其中一个包含该图所有边的一半，另一个对应的割也包含所有边的一半。我们考虑该性质的特殊情形，即其中一个子集是顶点覆盖，并将对应的图称为简单可平衡图。关于可平衡与简单可平衡正则图，我们证明了若干结果：首先，利用独立数条件刻画了简单可平衡正则图；其次，回应了Dailly、Hansberg和Ventura于2021年提出的问题，证明每个三次图都是可平衡的；第三，应用Brooks定理，证明每个阶数能被4整除的4-正则图是可平衡的；最后，证明判定9-正则图是否为简单可平衡的问题是NP完全的。