# gr-t3 - NAME TEST3/MAD3305 Page 1 of 4 General directions...

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NAME: TEST3/MAD3305 Page 1 of 4 _________________________________________________________________ General directions: Read each problem carefully and do exactly what is requested. Show all your work neatly. Use complete sentences and use notation correctly. Make your arguments and proofs as complete as possible. Remember that what is illegible or incomprehensible is worthless. _________________________________________________________________ 1. (15 pts.) (a) If G is a connected planar graph with 6 vertices, what can you tell me about the size of G? (b) For which pairs of integers r and s is K r,s planar and for which is K r,s nonplanar? Provide a brief explanation and/or a plane graph drawing, as appropriate, to deal with the various situations. _________________________________________________________________ 2. (10 pts.) (a) Suppose that G is a bipartite graph with partite sets U and W with U W . What does it mean to say that U is neighborly? (b) Recall that your friendly n-cubes are defined recursively by Q 1 = K 2 , and for n 2, Q n = Q n-1 x K 2 . Do these friendly bipartite graphs have perfect matchings?? Explain briefly.

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TEST3/MAD3305 Page 2 of 4 _________________________________________________________________ 3. (15 pts.) (a) Show that the graph below has a strong
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