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 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.

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， 103*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≤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.

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 least six, then we would have 2e= nvd

Vv6)(

.

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.

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

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.

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.

G

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.

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.

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.

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.

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

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.

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

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

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.

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 eE, then

PG(k) =PGe(k)-PGe(k)

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).

Exercise: P324 14,15,26,27 1.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.

2.Let G be a connected planar graph with n≥3 and f<12. Then G has a region with the degree less than 5.

3.Prove corollary 5.1 4.Prove figure 1 is a non planar graph 5.In figure 2, find these values (G), *(G), ’(G).

figure 1 figure 2