gr-t3

# 15 pts a what is a clique b sketch the shadow graph

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5. (15 pts.) (a) What is a clique? (b) Sketch the shadow graph S(C 5 ) of a generic 5-cycle below. What is χ (S(C 5 ))?? (c) How is the Grötzsch graph, which we will denote by G here, obtained from the shadow graph of Part (b) above?? It turns out that ω (G) = 2 and χ (G) = 4. What is the significance of this?? _________________________________________________________________ 6. (10 pts.) Prove, by induction on the size of the graph, that if G is a connected plane graph of order n, size m, and having r regions, thenn-m+r=2

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TEST3/MAD3305 Page 4 of 4 _________________________________________________________________ 7. (15 pts.) (a) What is a legal (or feasible) flow in a networkN?[ Hint: Definition. ] (b) Obtain a maximum flow f in the network below, and verify the flow is a maximum by producing a set of vertices S that produces a minimum cut. Check that the total capacity of that cut is the same as the value of your max flow. _________________________________________________________________ 8. (10 pts.) Which complete bipartite graphs K r,s are Hamiltonian and which are not? Explain briefly. [Hint: When can you use Dirac? What is the well-known necessary condition?]
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15 pts a What is a clique b Sketch the shadow graph SC 5 of...

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