Investigate!

For the complete graphs \(K_n\text{,}\) we would like to be able to say something about the number of vertices, edges, and (if the graph is planar) faces. Let's first consider \(K_3\text{:}\)

  1. How many vertices does \(K_3\) have? How many edges?

  2. If \(K_3\) is planar, how many faces should it have?

Repeat parts (1) and (2) for \(K_4\text{,}\) \(K_5\text{,}\) and \(K_{23}\text{.}\)

What about complete bipartite graphs? How many vertices, edges, and faces (if it were planar) does \(K_{7,4}\) have? For which values of \(m\) and \(n\) are \(K_n\) and \(K_{m,n}\) planar?

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