5. (15 pts.)
(a)
What is a clique?
(b)
Sketch the shadow graph S(C
5
) of a generic 5cycle 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, thennm+r=2
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_________________________________________________________________
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 wellknown necessary condition?]
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 Summer '12
 Rittered
 Graph Theory, Planar graph, Kvertexconnected graph, Complete bipartite graph

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