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

• Homework Help
• salammath
• 1

This preview shows page 1 out of 1 page.

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.
• Spring '13
• Dr.ZAre
• Graph coloring, Prove, simple planar graph, color critical graph, Hamiltonian bipartite graph

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern