  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.
