AMS/MBA 546 (Fall, 2010)
Estie Arkin
Network Flows
Homework Set # 9 (last)
Due Thursday, December 9, 2010.
1). Stable marriages with an unequal number of men and women: Let
W
be the set of women and
M
be
the set of men, and assume that

W

<

M

. A matching is unstable if there is a man
m
and woman
w
such
that:
•
m
and
w
are not partners in the matching;
•
m
is either unmarried, or prefers
w
to his wife in the matching;
•
w
is either unmarried, or prefers
m
to her husband in the matching.
Prove that a stable marriage always exists.
2). Given an undirected graph
G
= (
N, E
), recall that
χ
(
G
) is the (node) colouring number of
G
(the fewest
colours needed to colour all nodes of
G
), and
α
(
G
) is the size of the largest independent (stable) set (subset
of nodes no two of which are connected by an edge). The following parts are independet of each other.
(a). Show that
χ
(
G
)
·
α
(
G
)
≥
n
where
n
is the number of nodes in
G
.
(b). Let
G
be a tree. Give a polynomial time algorithm for computing
α
(
G
).
(c). Let
G
be a planar graph. Give a polynomial time approximation algorithm that returns an independent
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 Fall '08
 Arkin,E
 Graph Theory, Planar graph, Estie Arkin, maximum planar subgraph

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