Graph coloring

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1. Graph Coloring By M . Baranitharan Computer Science and Engineering

2. What is Graph Coloring? Graph Coloring is an assignment of colors (or any distinct marks) to the vertices of a graph. Strictly speaking, a coloring is a proper coloring if no two adjacent vertices have the same color

3. Origin of Problem

4. Why Graph Coloring? Many problems can be formulated as a graph coloring problem including Time Tabling, Scheduling, Register Allocation, Channel Assignment A lot of research has been done in this area so much is already known about the problem space.

5. Origin Coloring theory started with the problem of coloring the countries of a map in such a way that no two countries that have a common border receive the same color. If we denote the countries by points in the plane and connect each pair of points that correspond to countries with a common border by a curve, we obtain a planar graph.

6. Map of the countries

7. Assume

8. Vertex Coloring Vertex Coloring: It is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color.

9. Edges Coloring An edge coloring assigns a color to each edge so that no two adjacent edges share the same color .

10. Face Coloring A face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.

11. Chromatic number The chromatic number of a graph is the minimum number of colors in a proper coloring of that graph. If chromatic number is r then the graph is r-chromatic. Chromatic number: 4

12. Chromatic Polynomial Polynomial which gives the number of ways of proper coloring a graph using a given number of colors Ci = no. of ways to properly color a graph using exactly i colors λ = total no of colors λ C i = selecting I colors out of λ colors Σ C i λ C i = total number of ways a graph can be properly colored using λ or lesser no. of colors P n ( λ ) of G = Σ C i λ C i

13. Applications

14. Thank you