{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Combinatorics2008aug

# Combinatorics2008aug - Qualifying Exam in Combinatorics 19...

This preview shows page 1. Sign up to view the full content.

Qualifying Exam in Combinatorics 19 August, 2008 1. Let Γ be a 3-connected planar graph with planar dual Γ * . Prove or disprove: If Γ is a Cayley graph and Γ * is is vertex-transitive, then Γ * is a Cayley graph. If you believe the statement to be true, give a proof, or at least discuss what must be done to give a correct proof. If you believe the statement to be false, then give a counter-example, proving that exactly one of Γ and Γ * is a Cayley graph. 2. (a) Starting with the definition of derangement , derive an explicit formula for the n th derangement number D n . (Assume D 0 = 1 and D 1 = 0.) (b) Obtain an exponential generating function for the sequence { D n : n 0 } . (c) Give a combinatorial proof (not an algebaic proof) of the recurrence D n = ( n - 1)( D n - 1 + D n - 2 ) ( n 2) . 3. Describe how to “complete” an affine plane of order n to obtain a projective plane of order n . Then explain what this has to do with
This is the end of the preview. Sign up to access the rest of the document.

{[ 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