Theorem 6.5 provides a key insight into determining the balanceability of regular graphs with degrees greater than or equal to four (k ≥ 4). Specifically, it states that every 4-regular graph of order n, where n is divisible by 4, is balanceable. This theorem establishes a necessary and sufficient condition for balanceability in this specific case. By ensuring that the order of the graph is divisible by 4, one can conclude that such a graph will be balanceable.
Furthermore, Lemma 6.2 highlights that any connected 4-regular graph of odd order cannot be balanceable. This information helps in ruling out certain cases where the graph would not exhibit balanceability due to its odd order.
In summary, when dealing with k≥4 regular graphs, one should first check if the order of the graph is divisible by four as a necessary condition for potential balanceability. Additionally, considering whether the graph is connected or not plays a crucial role in determining its balanceability status based on established results like Theorem 6.5 and Lemma 6.2.