91Ó°ÊÓ

Show that every graph can be embedded in \(\mathbb{R}^{3}\) with all edges straight.

Find an Euler formula for disconnected graphs.

Show that every connected planar graph with \(n\) vertices, \(m\) edges and finite girth \(g\) satisfies \(m \leqslant \frac{g}{g-2}(n-2)\).

A graph is called outerplanar if it has a drawing in which every vertex lies on the boundary of the outer face. Show that a graph is outerplanar if and only if it contains neither \(K^{4}\) nor \(K_{2,3}\) as a minor.

Show that a 2-connected plane graph is bipartite if and only if every face is bounded by an even cycle.

\(^{+}\)Show that a connected plane multigraph has a plane dual.