The balancing number and list balancing number of some graph classes

Given a graph G, a 2-coloring of the edges of Kₙ is said to contain a copy of G if we can find a copy of G such that half of its edges in each color class. If there exists an integer k such that, for n large, every 2-coloring of Kₙ with more than k edges in each contains a balanced copy of G, then we say that G is balanceable. The integer k such that this holds is called the balancing number of. In this paper, we define a more general variant of the balancing number, the balancing number, by considering 2-list edge colorings of Kₙ, where edge e has an associated list $L(e)$ which is a nonempty subset of the set \,b\\. In this case, edges e with L(e) = \,b\\ act as in the sense that their color can be chosen r or b as needed. In to the balancing number, every graph has a list balancing number., if the balancing number exists, then it coincides with the list number. We give the exact value of the list balancing number for all cycles except $4k$-cycles for which we give tight bounds. In addition, we give general for the list balancing number of non-balanceable graphs based on the number of its subgraphs, and study the list balancing number of K₅, turns out to be surprisingly large.