hw09 - Math 412 HW9 Due Friday Students in the three credit...

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Math 412 HW9 Due Friday, April 29, 2016 Students in the three credit hour course must solve five of the six problems. Students in the four credit hour course must solve all six problems. 1 . Prove that if G is a color critical graph, then the graph G 0 generated from it by applying Mycielski’s construction is also color critical. 2 . Alternate proof of Tur´ an’s Theorem, including uniqueness (a) Prove that a maximal simple graph with no ( r + 1)-clique has an r -clique. (b) Prove that e ( T n,r ) = ( r 2 ) + ( n - r )( r - 1) + e ( T n - r,r ). (c) Use parts (a) and (b) to prove Tur´ an’s Theorem by induction on n , including the charac- terization of graphs achieving the bound. 3 . (a) Prove that χ ( C n ; k ) = ( k - 1) n + ( - 1) n ( k - 1) (b) For H = G K 1 , prove that χ ( F ; k ) = k · χ ( G ; k - 1). Using this and part (a), find the chromatic polynomial of the wheel C n K 1 . 4 . Let G be an n -vertex simple planar graph with girth k . Prove that G has at most ( n - 2) k k - 2 edges. Use this to prove that the Petersen graph is nonplanar.
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  • Spring '13
  • Dr.ZAre
  • Graph coloring, Prove, simple planar graph, color critical graph, Hamiltonian bipartite graph

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