Let
∅ 6
=
S
⊆
V
1
. Since
G
is
r
regular, the number of edges joining the
vertices of
S
and the vertices of
N
(
S
) is equal to
r

S

. All these edges are
incident with the vertices of
N
(
S
).
However, the total number of edges
incident with the vertices of
N
(
S
) is equal to
r

N
(
S
)

.
Hence
r

N
(
S
)
 ≥
r

S

.
Therefore,

N
(
S
)
 ≥ 
S

and by K¨
onigHall Theorem,
G
contains a
matching that matches all vertices in
V
1
.
Systems of distinct representatives
Definition 4.
A collection
A
1
, A
2
, . . . , A
n
,
n
≥
1, of finite nonempty sets
(not necessarily distinct) is said to have a
a system of distinct representatives
(SDR)
if there exists a set
{
a
1
, a
2
, . . . , a
n
}
of distinct elements such that
a
i
∈
A
i
for 1
≤
i
≤
n
.
Theorem 4
(Hall’s Theorem)
.
A collection
{
A
1
, A
2
, . . . , A
n
}
of finite nonempty
sets (not necessarily distinct) has a system of distinct representatives if and
only if for any 1
≤
k
≤
n
the union of any
k
of these sets contains at least
k
elements.
Proof.
Set
A
=
{
A
1
, A
2
, . . . , A
n
}
. Define the bipartite graph
G
with bipar
tition
{A
,
∪
n
i
=1
A
i
}
such that
A
i
is adjacent to
a
∈ ∪
n
i
=1
A
i
if and only if
a
∈
A
i
. Then
A
has an SDR if and only if
G
has a matching under which
all vertices
A
1
, A
2
, . . . , A
n
are matched, which, by K¨
onigHall Theorem, is
true if and only if for any
J
⊆ {
1
,
2
, . . . , n
}
,
 ∪
i
∈
J
A
i
 ≥ 
J

.
Remarks
•
K¨
onigHall Theorem is equivalent to Hall’s Theorem.
•
In the marriage problem, everyone can be married
⇔
every set of
k
women are collectively compatible with at least
k
men
⇔
every set of
k
men are collectively compatible with at least
k
women.
Augmentation along an augmenting path
Definition 5.
If
M
is a matching in
G
and
P
is an augmenting path with
respect to
M
, by swapping edges in and out of
M
in
P
we obtain a matching
M
0
with

M
0

=

M

+ 1, namely
M
0
= (
M

M
∩
E
(
P
))
∪
(
E
(
P
)

M
∩
E
(
P
))
.
This operation is called
augmentation
(more specifically, augmentation
of
M
along the augmenting path
P
).
•
All vertices on
P
are matched under
M
0
, whilst the two endvertices
of
P
are unmatched under
M
.
•
Most algorithms for finding maximum matchings are based on the
following idea: Keep augmenting until there is no augmenting path.
•
We will learn how to implement this systematically for bipartite graphs.
Alternating trees
Definition 6.
Let
M
be a matching in
G
and
u
an unmatched vertex with
respect to
M
. An
alternating tree
in
G
with respect to
M
and root
u
is a
tree
T
such that all (
u, v
)paths in
T
are alternating with respect to
M
.