 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，
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.
 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 G has
not any subgraph homeomorphic to K3.3 or
K5.
 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 kcolourable 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 scoloring 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 eE, 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