1. 5.9.2 Characterizations of Planar Graphs
1930
Kuratowski (库拉托斯基)
Two basic nonplanar graphs: K5 and K3,3

2. Definition 43: If a graph is planar, so will be any graph obtained by omitted an edge {u,v} and adding a new vertex w 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.

4. 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.

5. 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.

6. Example: Let G=(V,E)，|V|=7
Example: Let G=(V,E)，|V|=7. If G has a subgraph homeomorphic to K5, then it is a nonplanar graph, and is a planar graph.

7. 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.

10. 2. Region(face) colourings
Definitions 46: A edge of the graph is called a bridge, if the edge is not in any circuit. A connected planar graph is called a map, If the graph has not any bridge.
Definition 47: A proper region coloring of a map G is an assignment of colors to the region of G, one color to each region, such that adjacent regions receive different colors. An proper region coloring in which k colors are used is a k-region coloring. A map G is k-region colorable if there exists an s-coloring of G for some s  k. The minimum integer k for which G is k- region colorable is called the region chromatic number. We denoted by *(G). If *(G) = k, then G is k-region chromatic.

13. Four Colour Conjecture Every map (plane graph) is 4-region colourable.
Definition 48：Let G be a connected plane graph. Construct a dual Gd as follows:
1)Place a vertex in each region of G; this forms the vertex set of Gd.
2)Join two vertices of Gd by an edge for each edge common to the boundaries of the two corresponding regions of G.
3)Add a loop at a vertex v of Gd for each bridge that belongs to the corresponding region of G. Moreover, each edge of Gd is drawn to cross the associated edge of G, but no other edge of G or Gd.

15. Theorem 5.31 Every planar graph with no loop is 4-colourable if and only if its dual is 4-region colourable.

16. 3. Edge colorings
Definition 49:An proper edge coloring of a graph G is an assignment of colors to the edges of G, one color to each edge, such that adjacent edges receive different colors. An edge coloring in which k colors are used is a k-edge coloring. A graph G is k-edge colorable if there exists an s-edge coloring of G for some s k. The minimum integer k for which G is k-edge colorable is called the edge chromaticumber or the chromatic index ’(G) of G. If ’(G) = k, then G is k-edge chromatic.

18. 4. Chromatic polynomials
Definition 50: Let G =(V, E) be a simple graph. We let PG(k) denote the number of ways of proper coloring the vertices of G with k colors. PG will be called the chromatic function of G.
Example
For the graph G PG(k) =k (k-1)2

19. If G = (V, E ) with |V | = n and E =, then G consists of n isolated points, and by the product rule PG(k ) = k n.
If G =Kn, the complete graph on n vertices, then at least n colors must be available for a proper coloring of G. Here, by the product rule
P G(k ) = k (k-1)(k-2)...(k-n + 1).
We see that for k < n, P G(k ) = 0, which indicates there is no proper k -coloring of Kn

20. Let G = (V, E ) be a simple connected graph
Let G = (V, E ) be a simple connected graph. For e = {a, b}E, let Ge denote the subgraph of G obtained by deleting e from G, without removing the vertices a and b. Let Ge be the quotient graph of G obtained by merging the end points of e.
Example: Figure below shows the graphs Ge and Ge for the graph G with the edge e as specified.

21. Theorem 5.31 Decomposition Theorem for Chromatic Polynomials (色多项式分解定理) : If G = (V, E) is a connected graph and eE, then
PG(k) =PGe(k)-PGe(k)

23. Suppose that a graph is not connected and G1 and G2 are two components of G.
Theorem 5.32: If G is a disconnected graph with G1,G2,…Gw, then PG(k)=PG1(k)PG2(k)…PGw(k).
Next: Abstract algebra, Operations on the set 9.1, P330
Semigroups,monoids and groups 9.2 P341,9.4 p347

24. Exercise: P324 14,15,26,27
1.In figure 1, find these values  (G), *(G), ’(G).
figure 1