Ordering Non-Planar Graphs by Minimum Crossings

The question asks to arrange the given non-planar graphs in ascending order based on their minimum number of crossings. We need to find the crossing number, denoted as $cr(G)$, for each graph.

Minimum Crossing Numbers

The crossing number $cr(G)$ of a graph $G$ is the minimum number of edge crossings in any drawing of $G$ in the plane.

• Graph A: $K_6$ (Complete graph on 6 vertices)
  • The minimum number of crossings for $K_6$ is $cr(K_6) = 3$.
• Graph B: $K_7$ (Complete graph on 7 vertices)
  • The minimum number of crossings for $K_7$ is $cr(K_7) = 7$.
• Graph C: $K_{4,4}$ (Complete bipartite graph with partitions of size 4)
  • The minimum number of crossings for $K_{4,4}$ is $cr(K_{4,4}) = 4$.
• Graph D: $K_{5,5}$ (Complete bipartite graph with partitions of size 5)
  • The minimum number of crossings for $K_{5,5}$ is $cr(K_{5,5}) = 8$.

Ascending Order Calculation

We list the crossing numbers found:

• $cr(K_6) = 3$
• $cr(K_{4,4}) = 4$
• $cr(K_7) = 7$
• $cr(K_{5,5}) = 8$

Arranging these values in ascending order:

$3 < 4 < 7 < 8$

This corresponds to the graphs in the order:

$K_6, K_{4,4}, K_7, K_{5,5}$

Represented by the labels A, C, B, D.

Correct Arrangement

The correct arrangement of the non-planar graphs in ascending order based on the minimum number of crossings is A, C, B, D.