# hw03 - Math 412 HW3 Due Friday Students in the three credit...

• Homework Help
• salammath
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Math 412 HW3 Due Friday, February 19, 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 . For k 2, prove that every k -regular bipartite graph has no cut-edge and construct a bipartite graph with all vertex degrees in { k, k + 1 } that has a cut-edge. 2 . Given a nonincreasing list d = ( d 1 , . . . , d n ) of nonnegative integers and 1 k n , let d ( k ) be obtained from d by deleting d k and subtracting 1 from the d k largest elements remaining in the list. Prove that d is graphic if and only if d ( k ) is graphic. (Hint: Mimic the proof of Havel–Hakimi Theorem.) 3 . Suppose that G is a graph and D is an orientation of G that is strongly connected. Prove that if G has an odd cycle then D has an odd (directed) cycle. (Hint: Consider each pair { v i , v i +1 } in an odd cycle ( v 1 , . . . , v k ) in G .) 4 . For every odd n , construct an n -vertex tournament in which every vertex is a king. Does there exist such a tournament with 4 vertices?
• Spring '13
• Dr.ZAre
• Prime number, Bipartite graph, Complete bipartite graph, credit hour course, k-regular bipartite graph

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