Beaty's Theorem

Beaty's Theorem states that the complete graph with six vertices (called \(K_6\)) can be decomposed into 15 edge-disjoint complete bipartite graphs, each connecting three vertices to the other three (called \(K_{3,3}\)). In simpler terms, it means you can break down a complex network of six points connected every way into smaller, balanced groups where each group links two sets of three points without overlapping edges. This reveals a unique way to partition such a network, useful in graph theory and combinatorics for understanding symmetry and structure.