MATH 239 Fall 2008
Assignment 8
Solutions
Total 20 marks
1. The purpose of this problem is to prove that every nonmaximum matching admits an aug
menting path.
Let
M
be a matching in a graph
G
. Let
N
be a matching which is larger than
M
. Let
H
be the subgraph of
G
whose edges are (
M
∪
N
)
\
(
M
∩
N
) and whose vertices are the set of
vertices of
G
.
(a) Show that every component of
H
consists of either a path or a cycle.
(b) Prove that every path in
H
is an alternating path of
G
with respect to
M
.
(c) Prove that at least one path in
H
is an augmenting path of
G
with respect to
M
.
Solution.
(a) (2 marks) Every vertex of
H
is incident to at most two edges of
M
∪
N
, since
M
and
N
are matchings. Therefore, every vertex of
H
has degree equal to either zero, one or two.
It follows that every component of
H
is a path or a cycle (where we consider a vertex of
degree zero to be a path of length zero).
(b) (2 marks) Every vertex of
H
of degree two is incident with an edge from each of
M
and
N
. Hence every path in
H
is an alternating path of
G
with respect to
M
, and an
alternating path with respect to
N
.
(c) (2 marks) Observe that (
M
∪
N
)
\
(
M
∩
N
) is equal to the set of edges which belong
to exactly one of
M
or
N
. Since
N
has more edges than
M
, the subgraph
H
has more
edges of
N
than of
M
.
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 Fall '09
 M.PEI
 Math, WI, Bipartite graph, g., maximum matching

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