Solution

The correct answer is A, C, B, D.

Key Points

• The problem involves arranging graphs based on the minimum number of edge crossings, and the given graphs are K₆, K₇, K_4,4, and K_5,5.
• The formula for calculating the minimum number of crossings for a complete graph K_n is given by:
  cr(K_n) = ⌊(1/4) * ⌊n/2⌋ * ⌊(n-1)/2⌋ * ⌊(n-2)/2⌋ * ⌊(n-3)/2⌋ ⌋.
• For a complete bipartite graph K_m,n, the formula for the minimum number of crossings is:
  cr(K_m,n) = ⌊(1/2) * ⌊m⌋ * ⌊m-1⌋ * ⌊n⌋ * ⌊n-1⌋ / (m + n - 2) ⌋.
• Using these formulas, we can calculate and compare the minimum number of crossings for each graph and arrange them in ascending order.

Step-by-Step Solution

• Step 1: Calculate cr(K₆):
  • Using the formula for complete graphs: cr(K₆) = ⌊(1/4) * 3 * 2 * 2 * 1 ⌋ = ⌊3⌋ = 3.
• Step 2: Calculate cr(K₇):
  • Using the formula for complete graphs: cr(K₇) = ⌊(1/4) * 3 * 3 * 2 * 1 ⌋ = ⌊4.5⌋ = 4.
• Step 3: Calculate cr(K_4,4):
  • Using the formula for complete bipartite graphs: cr(K_4,4) = ⌊(1/2) * 4 * 3 * 4 * 3 / (4 + 4 - 2) ⌋ = ⌊72 / 6⌋ = 12.
• Step 4: Calculate cr(K_5,5):
  • Using the formula for complete bipartite graphs: cr(K_5,5) = ⌊(1/2) * 5 * 4 * 5 * 4 / (5 + 5 - 2) ⌋ = ⌊200 / 8⌋ = 25.
• Step 5: Arrange the graphs in ascending order:
  • From the calculations: cr(K₆) = 3, cr(K₇) = 4, cr(K_4,4) = 12, cr(K_5,5) = 25.
  • Ascending order: K₆, K_4,4, K₇, K_5,5, which corresponds to the order A, C, B, D.

Additional Information

• Planar Graphs:
  • A graph is planar if it can be drawn on a plane without any edges crossing.
  • Examples of planar graphs include K₄ and K_3,3 under specific conditions.
• Non-Planar Graphs:
  • Graphs like K₅, K₆, and K_4,4 are non-planar due to their high edge density.
• Applications of Graph Theory:
  • Used in network design, computer science, and optimization problems.
  • Graph crossings are important in VLSI design and circuit layout problems.