392
Solutions
to
Exercises
(b)
After the first night, we remove the edges corresponding to the dates which occurred.
We
are left
with a bipartite graph in which every vertex has degree 4. Again, Problem 6 guarantees a perfect
matching. Continue for the
five
nights, reducing the degree by 1 each time.
6. (a) [BB] Bruce
+t
Maurice, Edgar
+t
Michael, Eric
+t
Roland, Herb
+t
Richard.
(b)
If
Roland and Bruce share a canoe, then Maurice and Michael would both have to be with Edgar.
(c) No.
If
Bruce and Roland were together, then both Maurice and Michael would have to be in the
same canoe with Edgar. On the other hand,
if
Bruce and Roland were not in the same canoe, then
Bruce and Roland would have to be in the same canoe with Maurice and Michael (in some order),
leaving no one for Edgar since he refuses to
be
with Herb.
7.
(a) [BB] Construct a bipartite graph with vertex sets
VI
and
V2,
where
VI
has
n
vertices corresponding
to
AI,
...
,
An,
V2
has one vertex for each element
of
S
and there is an edge joining
Ai
to s
if
and
only
if
s E
Ai.
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 Summer '10
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 Graph Theory, Necessary and sufficient condition, Bipartite graph, Roland

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