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CS 594 Graph Theory, Spring 2014 Homework 1 1. For each of the following, either draw a graph with the given degree sequence or prove the sequence is not graphic. (See pp. 44-45, especially theorem 1.3.31.) a. (5,5,5,3,2,2) b. (5,5,5,3,3,3,2,2) c. (7,3,3,3,3,3,3,1) d. (5,5,5,4,1,1,1,1,1) 2. Prove that every graph has at least two vertices with the same degree. 3. Prove that every set of six people contains at least three mutual acquaintances or three mutual strangers. 4. Prove that if ݒis a cut vertex of ܩ, then ܩҧ െ ݒis connected. 5. Prove that if G has no isolated vertex and no induced subgraph with exactly two edges, then G is a complete graph. 6. Prove that if G is a connected graph not having P4 or C3 as an induced subgraph, then G is a complete bipartite graph. 7. How many cycles of length n are in Kn?