)=2a
n
d
∆
(C
4
) = 2, we need
at least one vertex.
There is not a 3regular
graph of order 5. So
6 vertices will be the
best that we can do.
_________________________________________________________________
8. (10 pts.)
Prove exactly one of the following propositions.
Indicate clearly which you are demonstrating.
(a)
If G is a nontrivial graph, then there are distinct
vertices u and v in G with deg(u) = deg(v).
(b)
If G is a graph of order n and deg(u) + deg(v)
≥
n1f
o
r
each pair of nonadjacent vertices u and v, then G is connected.
// Review??
(a): Theorem 2.14, page 51.
(b): Theorem 2.4, page 34.
Proofs were also done in class.
These varied somewhat from those
of the text.
_________________________________________________________________
9. (10 pts.)
(a) Suppose G is a bipartite graph of order at
least 5.
Prove that the complement of G is not bipartite. [Hint:
At least one partite set has three elements. Connect the dots?]
Suppose that G is a bipartite graph of order at least 5 with
partite sets U and W.
At least one of U and W has at least 3
elements.
Suppose without loss of generality,
W
≥
3.
Label
three of the members of W with u,v, and w.
Since these vertices
are in the same partite set of G, none of these three vertices is
adjacent to any other of the three.
Thus,
uv
,
vw
,
uw
∈
E
(
G
)
⇒
G contains a
3
cycle
.
Thus, the complement of G is not bipartite. [Problem 1.25?]
(b) Display a bipartite graph G of order 4 and its bipartite
complement.
Label each appropriately and give partite sets for
each bipartite graph.
//There are a multitude of examples.
See me if you need help
with this!
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 Summer '12
 Rittered
 Graph Theory, Vertex, subgraph

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