Final answer:
K2,3 is nonplanar, every embedding of a planar graph does not have edge crossings, C20 is a planar graph, and K5 does not have a planar embedding.
Step-by-step explanation:
(a) To prove that K2,3 is nonplanar, we can use Euler's formula, which states that for any planar graph with V vertices, E edges, and F faces, V - E + F = 2. K2,3 has 5 vertices and 5 edges, so we substitute these values into the formula: 5 - 5 + F = 2, which simplifies to F = 2. This means that K2,3 has 2 faces, which is not possible for a planar graph. Therefore, K2,3 is nonplanar.
(b) Every embedding of a planar graph does not have edge crossings. This statement is true. In a planar graph, the edges are drawn on a plane in such a way that they do not intersect. This ensures that there are no edge crossings in any embedding of a planar graph.
(c) To determine if C20 is a planar graph, we can use Euler's formula. C20 has 20 vertices and 20 edges. Substituting these values into the formula: 20 - 20 + F = 2, which simplifies to F = 2. This means that C20 has 2 faces, satisfying Euler's formula for a planar graph. Therefore, C20 is a planar graph.
(d) To determine if K5 has a planar embedding, we can again use Euler's formula. K5 has 5 vertices and 10 edges. Substituting these values into the formula: 5 - 10 + F = 2, which simplifies to F = 7. This means that K5 has 7 faces, which is not possible for a planar graph. Therefore, K5 does not have a planar embedding.