Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 457 — #465
12.5. Bipartite Graphs & Matchings 457
neighbors of some set S of vertices is the image of S under the edge-relation, that
is,
N.S/ WWD f r j hs—ri 2 E.G/ for some s 2 S g:
S is called a bottleneck if
jSj > j N.S/j:
Theorem 12.5.4 (Hall’s Theorem). Let G be a bipartite graph. There is a matching
in G that covers L.G/ iff no subset of L.G/ is a bottleneck.
An Easy Matching Condition
The bipartite matching condition requires that every subset of men has a certain
property. In general, verifying that every subset has some property, even if it’s easy
to check any particular subset for the property, quickly becomes overwhelming
because the number of subsets of even relatively small sets is enormous—over a
billion subsets for a set of size 30. However, there is a simple property of vertex
degrees in a bipartite graph that guarantees the existence of a matching. Call a
bipartite graph degree-constrained if vertex degrees on the left are at least as large
as those on the right. More precisely,
Definition 12.5.5. A bipartite graph G is degree-constrained when deg.l/ deg.r/
for every l 2 L.G/ and r 2 R.G/.
For example, the graph in Figure 12.9 is degree-constrained since every node on
the left is adjacent to at least two nodes on the right while every node on the right
is adjacent to at most two nodes on the left.
Theorem 12.5.6. If G is a degree-constrained bipartite graph, then there is a
matching that covers L.G/.
Proof. We will show that G satisfies Hall’s condition, namely, if S is an arbitrary
subset of L.G/, then
j N.S/j jSj: (12.2)
Since G is degree-constrained, there is a d > 0 such that deg.l/ d deg.r/
for every l 2 L and r 2 R. Since every edge with an endpoint in S has its other
endpoint in N.S/ by definition, and every node in N.S/ is incident to at most d
edges, we know that
dj N.S/j #edges with an endpoint in S:
Also, since every node in S is the endpoint of at least d edges,
#edges incident to a vertex in S djSj: