MATH 443 Problems #5 Due Thursday Nov 14. These problems are for classroom presentations and some fraction of them may be final exam questions. 30. Let G be a planar graph with girth(G) = k. Show that |E(G)| ≤ k(|V (G)|−2) k−2 31. Let x1 , x2 , . . . , xn be a set of points in the plane R2 . Assume all points are at distance at least one. Show that the number of pairs of points at distance exactly one is at most 3n − 6. 32. For the cycle Cn show that χ(Cn ; k) = (k − 1)n + (−1)n (k − 1). Use this to compute χ(Cn ). 33. Let G be a connected simple graph. Let G have a cut vertex v so that G can be thought of as the union of two connected subgraphs H, K which overlap on the single vertex v. Show that χ(G; k) = 1 χ(H; k)χ(K; k) k 34. Let G be a graph for which every pair of odd cycles has at least one vertex in common. Prove that χ(G) ≤ 5. Hint: if G has a triangle, would you have a proof?