Use the HavelHakimi Theorem to show that the
sequence
s:
5,5,4,4,3,3
is the degree sequence of some graph. Then show there is a plane
graph with this sequence as its degree sequence. [You may need
to do a second drawing.]
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4. (10 pts.)
(a) Explain why
is Hamiltonian for
n
≥
5.
C
n
(b) Give an example of a simple connected graph G of order n
with
∆
(G) < n/2 that is Hamiltonian. What does this tell us
about Ore’s Theorem?
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5. (15 pts.)
Each of the following propositions may be proved
by induction. Prove exactly one of them after clearly indicating
which you are proving.
(a) Every nontrivial connected graph G has a spanning tree.
[Hint: If the order of the graph G is at least 3, Theorem 1.10
implies that G has a vertex v with G  v connected.]
(b) If G is a connected plane graph of order n, size m, and
having r regions, then n  m + r = 2. [Hint: Do induction on the
size of G.]
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6. (10 pts.)
(a) What can you say about the order of any
nontrivial digraph in which no two vertices have the same
outdegree, but every two vertices have the same indegree? Why??
(b) What degree condition characterizes Eulerian digraphs? Is
there an Eulerian digraph whose underlying graph is not Eulerian?
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 Summer '12
 Rittered
 Graph Theory, Planar graph, 5 pts, outdegree, D. König

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