5.8 Graph Matching
 Example: Set of worker assign to a
set of task
 Four tasks are to be assigned to four
workers.
 – Worker 1 is qualified to do tasks B
and C
 – Worker 2 is qualified to do tasks
A,C and D
 – Worker 3 is qualified to do tasks B
and D
 – Worker 4 is qualified to do task A
and C.
 Can all 4 workers be assigned to
different tasks for which they are
qualified?
 Example 2: The Marriage Problem:Given a set of men, each
of whom knows some women from a given set of women,
under what conditions is it possible for all men to marry
women they know?
 Four men each know some of four women
 Peter knows Mary and Ann
 Kevin knows Mary, Ann, Rose and Tina
 Brian knows Mary and Ann
 Fred knows Ann
 Is it possible for all the men to marry women they know?
 Graph Matching
 Definition 36: A matching M in a graph G(V;E) is a
subset of the edge set E such that no two edges in M
are incident on the same vertex. The size of a
matching M is the number of edges in M. For a
graph G(V;E), a matching of maximum size is
called a maximum matching.
M1={e1,e7},M2={e1,e2,e5},M3
={e1,e2,e5,e6}and M4=
{e1,e2,e7,e8} are matching.
M3 and M4 are maximum
matching.
 Definition 37: If M is a matching in a graph
G, a vertex v is said to be M-saturated if
there is an edge in M incident on v. Vertex v
is said to be M-unsaturated if there is no edge
in M incident on v.
M1={e1,e7}, M3={e1,e2,e5,e6}
M1-saturated : v
M1-unsaturated: u
M3-saturated:u,v
M={e1,e3,e7,e8},
M1={e1,e3,e5,e6}
M1 and M are perfect matching.
 Definition 38: A matching M of G
is perfect if all vertices of G are
M-saturated. If G(V1;V2) is a
bipartite graph then a matching
M of G that saturates all the
vertices in V1 is called a complete
matching from V1 to V2.
 M={{v1,u1},{v2,u2},{v3,u3}}
 M is a complete matching from V1 to V2, but it is
not a complete matching from V2 to V1.
 Definition 39: Given a matching M in a graph
G, a M-alternating path (cycle) is a path (cycle)
in G whose edges are alternately in M and
outside of M (i.e. if an edge of the path is in M,
the next edge is outside M and vice versa). A Malternating path whose end vertices are Munsaturated is called an M-augmenting path.
 Theorem 5.25：M is a maximal matching of G
iff there is no augmenting path relative to M.
 Proof: (1) There is no augmenting path relative
to M, we prove M is a maximal matching of G .
 Suppose M and N are matching with |M|<|N|.
 To see that MN contains an augmenting path
relative to M we consider G’ = (V’, MN )
 1≤dG’(v)≤2
 Since |M|<|N|, MN has more edges from N
than M and hence has at least one augmenting
path relative to M
 contradiction
 (2)If M is a maximal matching of G then there is
no augmenting path relative to M
 Assume that there exists an M-augmenting path
p.
 To see that Mp is a matching of G and
|Mp|>|M|
 1)Mp is a matching of G
 e1,e2Mp, e1 and e2 are not adjacent.
 2) |Mp|>|M|
 |Mp|=|(M∪p)-(M∩p)|= |(M∪p)|-|(M∩p)|
 =|M|+|p|-2|(M∩p)|
 =|M|+1
 Definition 40: Given a bipartite graph
G(V1;V2), and a subset of vertices S V,
the neighborhood N(S) is the subset of
vertices of V that are adjacent to some
vertex in S, i.e.
 N(S) ={vV|uS,{u,v}E(G)}
A={v1,v3},N(A)={v2,v6,v4}
A1={v1,v4},N(A1)={v2,v6,v4,v3,
v5,v1}
 Theorem 5.26: Let G(V1,V2) be a bipartite graph
with |V1|=|V2|. Then a complete matching of G
from V1 to V2 is a perfect matching
 Theorem 5.27 (Hall's Theorem) Let G(V1; V2) be a
bipartite graph with |V1|≤|V2|. Then G has a complete
matching saturating every vertex of V1 iff |S|≤|N(S)|
for every subset SV1
 Example: Let G be a k-regular bipartite graph. Then
there exists a perfect matching of G, where k>0.
 k-regular
 For AV1,E1={e|e incident a vertex of A}, E2={e|e
incident a vertex of N(A)}
 For eE1, eE2. Thus E1E2. Therefore |E1|≤|E2|.
 Because k|A|=|E1|≤|E2|=k|N(A)|, |N(A)|≥|A|.
 By Hall’s theorem, G has a complete matching M
from V1 to V2.
 Because |V1|=|V2|, Thus M is a perfect matching.
5.9 Planar Graphs
 5.9.1 Euler’s Formula
 Definitions 41: Intuitively, a graph G is
planar if it can be embedded in the plane,
that is, if it can be drawn in the plane
without any two edges crossing each other.
If a graph is embedded in the plane, it is
called a planar graph.
 Definition 42:A planar embedded of a
graph splits the plane into connected
regions, including an unbounded region.
The unbounded region is called outside
region, the other regions are called inside
regions.
 Theorem 5.28(Euler’s formula) If G is a
connected plane graph with n vertices, e edges
and f regions, then n -e+f= 2.
 Proof. Induction on e, the case e = 0 being as in
this case n = 1, e = 0 and f =1
 n-e+f=1-0+1=2
 Assume the result is true for all connected plane
graphs with fewer than e edges,
 e ≥ 1, and suppose G has e edges.
 If G is a tree, then n =e+1 and f= 1, so the result
holds.
If G is not a tree, let a be an
edge of a cycle of G and
consider G-a.
Clearly, G-a is a connected plane
graph with n vertices, e-1 edges
and f-1 regions,
so by the induction hypothesis, n(e-1) + (f- 1) = 2, from which it
follows that n -e +f = 2.
 Corollary 5.1 If G is a plane graph with n vertices, e
edges, k components and f regions, then n-e +f= 1+k.
 Corollary 5.2: If G is a connected planar simple
graph with e edges and n vertices where n ≥ 3, then
e≤3n-6.
Proof: A connected planar simple graph drawn in the
plane divides the plane into regions, say f of them.
The degree of each region is at least three(Since the graphs
discussed here are simple graphs, no multiple edges that
could produce regions of degree two, or loops that could
produce regions of degree one, are permitted).
The degree of a region is defined to be number of edges on
the boundary of this region.
We denoted the sum of the degree of the regions by s.
 Suppose that K5 is a planar graph, by
the Corollary 5.2,





n=5,e=10，
103*5-6=9, contradiction
K3,3，n=6,e=9,
3n-6=3*6-6=12>9=e，
But K3,3 is a nonplanar graph
 Corollary 5.3: If a connected planar simple
graph G has e edges and n vertices with n ≥
3 and no circuits of length three, then e≤2n4.
 Proof: Now, if the length of every cycle of G
is at least 4, then every region of (the plane
embodied of) G is bounded by at least 4
edges.
 K3,3 is a nonplanar graph
 Proof: Because K3,3 is a bipartite graph, it is
no odd simple circule.
Corollary 5.4:Every connected planar simple
graph contains a vertex of degree at most five.
Proof：If n≤2 the result is trivial
For n≥3, if the degree of every vertex were at
d (v )  6n .
least six, then we would have 2e= v
V
By the Corollary 5.2, we would have 2e≤6n-12.
contradiction.
 Corollary 5.5: Every connected planar
simple graph contains at least three vertices
of degree at most five, where n≥3.
 Next: Characterizations of Planar Graphs

Graph Colourings P320
 Exercise:
 1. Let G be a bipartite graph. Then G has a perfect matching
iff |N(A)|≥|A| for AV.
 2.Suppose that G is a planar simple graph. If the number of
edges of G less than 30, then there exists a vertex so that its
degree less than 5.
 3.Let G be a connected planar graph with δ(G)≥3 and f<12.
Then G has a region with the degree less than 5.
 4.Prove corollary 5.1
 5.Prove figure 1 is a nonplanar graph

figure 1