Question: 5 Let G be a graph equipped with a drawing without edge crossings where all vertices 15 pts of G lie at the boundary of

5 Let G be a graph equipped with a drawing without edge crossings where all vertices 15 pts of G lie at the boundary of the outer face of the drawing. We further assume the following conditions on the graph G and the drawing: (a) There are at least two faces in the drawing. (b) Every inner face is bounded by a cycle of length 3. (c) Every edge of G is contained in a cycle. The following figure is one example of such graphs. Figure 2: Under the assumptions above, Solve the following problems. (1) (5 pts) Show that x(G) 2 3. (2) (5 pts) Let H be a graph such that V(H) is the set of the inner faces and fif2 is an edge of H if and only if they share an edge. Prove that there is an inner face f such that dega(f) = 1. (Hint: You might want to use the average of degrees in H.) (3) (5 pts) Using (2) and mathematical induction on the size [V(G)|, prove that x(G)

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