The Balancing Number and Generalized Balancing Number of Some Graph Classes

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