Graph Theory
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A graph consists of a set of vertices and edges connecting pairs of vertices. Graph coloring assigns colors to vertices such that no adjacent vertices share the same color. The chromatic polynomial counts the number of valid colorings of a graph using a given number of colors. It was introduced to study the four color theorem and fundamental results were established in the early 20th century. The chromatic polynomial can be used to find the chromatic number of a graph.
Graph Theory
- 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
- 10. Examples: G = K4 P(G, x) = x(x-1)(x-2)(x-3) = x(4) G = Star5 P(G, x) = x(x-1)5 G = C4 P(G, x) = x(x-1)2 + x(x-1)(x-2)2 = x4 - 4x3 + 6x2 - 3x
- 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.
- 12. Chromatic Polynomials of complete graph (n-1) x(x-1)
- 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:-
- 14. Applications