The number of cycles of length 4 that are possible in the complete bipartite…

In a complete bipartite graph K(m,n) (vertex sets of size m and n, with every vertex on one side joined to every vertex on the other side, and no edges within a side), a cycle can only alternate between the two sides -- so any 4-cycle must use exactly 2 vertices from each side. For any 2 vertices chosen from the m-side and any 2 vertices chosen from the n-side, since a bipartite graph has no edge joining two vertices on the same side, the only way these 4 vertices can form a cycle is by alternating: a1-b1-a2-b2-a1. So each pair of vertices from one side combined with each pair from the other side gives exactly one distinct 4-cycle, and the total count of 4-cycles is C(m,2) x C(n,2).

Applying this to K(4,6), where m = 4 and n = 6:

1. Here m = 4 and n = 6, so the two vertex sets have 4 and 6 vertices respectively.

2. Number of ways to choose 2 vertices from the 4-vertex side: C(4,2) = (4 x 3)/(2 x 1) = 6.

3. Number of ways to choose 2 vertices from the 6-vertex side: C(6,2) = (6 x 5)/(2 x 1) = 15.

4. Each such pair from one side combined with each pair from the other side yields exactly one 4-cycle, so the total number of 4-cycles = C(4,2) x C(6,2) = 6 x 15 = 90.

Independent check: fix any 2 vertices {a1, a2} on the 4-side and any 2 vertices {b1, b2} on the 6-side. Since there is no edge between a1 and a2 (same side) nor between b1 and b2 (same side), the unique 4-cycle on these 4 vertices must alternate sides: a1-b1-a2-b2-a1. So the count C(4,2) x C(6,2) = 90 enumerates every 4-cycle exactly once -- no cycle is double-counted or missed.

Hence, the number of cycles of length 4 in K(4,6) is 90.