The Balancing Number and Generalized Balancing Number of Some Graph Classes

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 is 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 contains a balanced copy of $G$, then we say that $G$ is balanceable. The smallest integer $k$ such that this holds is called the balancing number of $G$.In this paper, we define a more general variant of the balancing number, the generalized balancing number, by considering 2-coverings of the edge set of $K_n$, where every edge $e$ has an associated list $L(e)$ which is a nonempty subset of the color set $\{r,b\}$. In this case, edges $e$ with $L(e) = \{r,b\}$ act as jokers in the sense that their color can be chosen $r$ or $b$ as needed. In contrast to the balancing number, every graph has a generalized balancing number. Moreover, if the balancing number exists, then it coincides with the generalized balancing number.We give the exact value of the generalized balancing number for all cycles except for cycles of length $4k$ for which we give tight bounds. In addition, we give general bounds for the generalized balancing number of non-balanceable graphs based on the extremal number of its subgraphs, and study the generalized balancing number of $K_5$, which turns out to be surprisingly large.