(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! This reveals why we want V(G)
≥
5 in (a).
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 Summer '12
 Rittered
 Graph Theory, Vertex, subgraph

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