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COMP 150-GT: Graph Theory Tufts University Prof. Lenore Cowen Fall 2007 HW 6: due Thursday,October 25 in class 1. Suppose the odd cycles in a graph G are pairwise intersecting, meaning that any two odd cycles share a common vertex. Prove that chi(G) ≤ 5. 2. Let n(G) denote the number of vertices of G. Prove that χ(G) + χG ≤ n(G) + 1. (Hint: use induction on n(G)). 3. Prove or disprove: χ(G) ≤ 1 + a(G) for every simple graph G, where a(G) is the average vertex degree of G (i.e. twice the number of edges divided by the number of vertices). 4. Use Brooks’ theorem to give a short proof of a special case of Vizing’s theorem: if G is a simple graph with ∆(G) = 3, then G is 4-edge colorable. 0 5. Prove that if G is regular and has a cut-vertex, then χ (G) > ∆.