1. Graphs
A graph G = (V, E) consists of a vertex set V
and an edge set E, where each edge is an
unordered pair of vertices.
2. Example
x e e6 2
z ee4 1 y e
w 3 e5
Vertex set V = {x,y,z,w}
Edge set
where e6
E =
is a
are
{ei: i=1,2,3,4,5,6}.
loop and
and 4 5
4
3. Colourings of graphs
A 3-colouring, satisfying the condition:
Every two adjacent vertices are assigned
colours.
different
4. Application of colourings
z1
y z2 2
y1
y2 y1
x1
x2 z1 x1
x2 z2
The traffic flow must 3 colours are required.
be separated into 3
periods.
5. Colourings of maps
In national map, neighbour provinces are
usually assigned different colours.
Zhejiang
Jiangxi
6. History of chromatic
polynomial
1.The chromatic polynomial was introduced by Birkhoff in 1912
as a way to attack the four-colour problem.
2. Whitney (1932) established many fundamental results.
3. Birkhoff and Lewis in 1946 conjectured that the chromatic of
any planar graph has no zeros larger than 4.
7. 4. R.C. Read in 1968 published an well
referenced introductory
polynomials.
article on chromatic
8. Chromatic
Polynomials
Chromatic Polynomials for a given
graph G, the number of ways of
coloring the vertices with x or fewer
colors is denoted by P(G, x) and is
called the chromatic polynomial of G
(in terms of x).
Examples:
G = chain of length n-1 (so there are n vertices)
P(G, x) = x(x-1)n-1
9. For any graph G of order n, if
n
P ( G , x ) = Σ a x i , i
i=1
11. Decomposition Theorem
To find chromatic number of a given graph
- no define algorithm so far
-Range can be found as follows
X(g)<= 1+Δ(g) , Δ(g) is the maximum degree of a vertex in graph.
13. Chromatic Polynomials through
Decomposition theorem
1. -Find a pair of non-adjacent
vertex.
2. Fuse(a,b) to from a simple graph
by replacing parallel edge with
single edge.
Repeat step 1 and 2 on these
graph till all nodes are comlete
graph
Examples:-
