1. N(S) ={vV|uS,{u,v}E(G)}
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}

2. Theorem 5. 26: Let G(V1,V2) be a bipartite graph with |V1|=|V2|
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

3. 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.

4. 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.

7. 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.

8. 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

9. e ≥ 1, and suppose G has e edges.
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 e be an edge of a cycle of G and consider G-e.
Clearly, G-e 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.

10. 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.

11. 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

12. 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≤2n-4.
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.

14. Corollary 5.5: Every connected planar simple graph contains at least three vertices of degree at most five, where n≥3.

15. 5.9.2 Characterizations of Planar Graphs
1930
Kuratowski (库拉托斯基)
Two basic nonplanar graphs: K5 and K3,3

16. Definition 43: If a graph is planar, so will be any graph obtained by omitted an edge {u,v} and adding a new vertex together with edges {u,w} and {w,v}. Such an operation is called an elementary subdivision.
Definition 44: The graphs G1=(V1,E1) and G2=(V2,E2) are called homeomorphic if they can be obtained from the same graph by a sequence of elementary subdivisions.

18. Theorem 5.29: (1)If G has a subgraph homeomorphic to Kn, then there exists at least n vertices with the degree more than or equal n-1.
(2) If G has a subgraph homeomorphic to Kn,n, then there exists at least 2n vertices with the degree more than or equal n.
Example: Let G=(V,E)，|V|=7. If G has a subgraph homeomorphic to K5, then has not any subgraph homeomorphic to K3.3 or K5.

19. Theorem 5. 30: Kuratowski’s Theorem (1930)
Theorem 5.30: Kuratowski’s Theorem (1930). A graph is planar if and only if it contains no subgraph that is homeomorphic of K5 or K3,3.
(1)If G is a planar graph, then it contains no subgraph that is homeomorphic of K5 , and it contains no subgraph that is homeomorphic of K3,3
(2)If a graph G does contains no subgraph that is homeomorphic of K5 and it contains no subgraph that is homeomorphic of K33 then G is a planar graph
(3)If a graph G contains a subgraphs that is homeomorphic of K5, then it is a nonplanar graph. If a graph G contains a subgraph that is homeomorphic of K3,3, then it is a nonplanar graph.
(4)If G is a nonplanar graph, then it contains a subgraph that is homeomorphic of K5 or K3,3.

20. 5.9.3 Graph Colourings 1.Vertex colourings
Definitions 45:A proper colouring of a graph G with no loop is an assignment of colours to the vertices of G, one colour to each vertex, such that adjacent vertices receive different colours. A proper colouring in which k colours are used is a k-colouring. A graph G is k-colourable if there exists a s-colouring of G for some s ≤ k. The minimum integer k for which G is k-colourable is called the chromatic number. We denoted by (G). If (G) = k, then G is k-chromatic.

22. NEXT:8.6 Colouring Graph P334 (Sixth) OR P320(Fifth)

23. 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