Graph colouring

Maulana Azad National Maulana Azad National InstituteInstitute of of TechnologyTechnology

Department of Computer Science & EngineeringDepartment of Computer Science & Engineering

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Graph ColoringGraph Coloring

Presented By:Presented By:

Priyank JainPriyank Jain

Shweta SaxenaShweta Saxena

What is Graph Coloring?What is Graph Coloring?

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

Origin of the problemOrigin of the problem

Why Graph Coloring?Why Graph Coloring?

Many problems can be formulated as a Many problems can be formulated as a graph coloring problem including Time graph coloring problem including Time Tabling, Tabling, Channel AssignmentChannel Assignment etc. etc.

A lot of research has been done in this A lot of research has been done in this area.area.

Channel AssignmentChannel Assignment

Find a channel assignment to R radio Find a channel assignment to R radio stations such that no station has a conflict stations such that no station has a conflict (there is a conflict if they are in vicinity)(there is a conflict if they are in vicinity)

Vertices – radio stations, edges – conflict, Vertices – radio stations, edges – conflict, colors – available channelscolors – available channels

TerminologyTerminology

K-ColoringK-Coloring A k-coloring of a graph G is a mapping of A k-coloring of a graph G is a mapping of

V(G) onto the integers 1..k such that adjacent V(G) onto the integers 1..k such that adjacent vertices map into different integers.vertices map into different integers.

A k-coloring partitions V(G) into k disjoint A k-coloring partitions V(G) into k disjoint subsets such that vertices from different subsets such that vertices from different subsets have different colors.subsets have different colors.


K-colorableK-colorable A graph G is k-colorable if it has a k-coloring.A graph G is k-colorable if it has a k-coloring.

Chromatic NumberChromatic Number The smallest integer k for which G is k-The smallest integer k for which G is k-

colorable is called the chromatic number of G.colorable is called the chromatic number of G.


K-chromatic graphK-chromatic graph A graph whose chromatic number is k is A graph whose chromatic number is k is

called a k-chromatic graph.called a k-chromatic graph.

ColoringColoring A coloring of a graph G assigns colors to the A coloring of a graph G assigns colors to the

vertices of G so that adjacent vertices are vertices of G so that adjacent vertices are given different colorsgiven different colors

ExampleExample

The chromatic number is four. Therefore this a 4-Chromatic Graph

ExampleExample

Problem: A state legislature has a Problem: A state legislature has a number of committees that meet each number of committees that meet each week for one hour. How can we schedule week for one hour. How can we schedule the committee meetings times such that the committee meetings times such that the least amount of time is used but such the least amount of time is used but such that two committees with overlapping that two committees with overlapping membership do not meet at the same membership do not meet at the same time.time.

Example (cont)Example (cont)

The chromatic number of this graph is four. Thus four hours suffice to schedule committee meetings without conflict.

An edge represents a conflict between to meetings

An vertex represents a meeting

Graph Colouring AlgorithmGraph Colouring Algorithm

There is no efficient algorithm available for There is no efficient algorithm available for coloring a graph with minimum number of coloring a graph with minimum number of colors.colors.

Graph coloring problem is a known NP Graph coloring problem is a known NP Complete problem. Complete problem.

NP Complete Problem NP Complete Problem

NP complete problems are problems NP complete problems are problems whose status is unknown. whose status is unknown.

No polynomial time algorithm has yet No polynomial time algorithm has yet been discovered for any NP complete been discovered for any NP complete problemproblem

It is not established that no polynomial-It is not established that no polynomial-time algorithm exist for any of them. time algorithm exist for any of them.

NP Complete ProblemNP Complete Problem

The interesting part is, if any one of the The interesting part is, if any one of the NP complete problems can be solved in NP complete problems can be solved in polynomial time, then all of them can be polynomial time, then all of them can be solved.solved.

Although Graph coloring problem is NP Although Graph coloring problem is NP Complete problem there are some Complete problem there are some approximate algorithms to solve the graph approximate algorithms to solve the graph coloring problem.coloring problem.

Basic Greedy AlgorithmBasic Greedy Algorithm

1.1. Color first vertex with first color.Color first vertex with first color.

2. Do following for remaining V-1 vertices. 2. Do following for remaining V-1 vertices. a) a) Consider the currently picked vertex Consider the currently picked vertex and color it with the lowest numbered and color it with the lowest numbered color that has not been used on any color that has not been used on any previously colored vertices adjacent to it. previously colored vertices adjacent to it. If all previously used colors appear on If all previously used colors appear on vertices adjacent to v, assign a new color vertices adjacent to v, assign a new color to it. to it.

Analysis of Greedy AlgorithmAnalysis of Greedy Algorithm

The above algorithm doesn’t always use The above algorithm doesn’t always use minimum number of colors. Also, the minimum number of colors. Also, the number of colors used sometime depend number of colors used sometime depend on the order in which vertices are on the order in which vertices are processedprocessed

Example:Example:

For example, consider the following two For example, consider the following two graphs. Note that in graph on right side, graphs. Note that in graph on right side, vertices 3 and 4 are swapped. If we vertices 3 and 4 are swapped. If we consider the vertices 0, 1, 2, 3, 4 in left consider the vertices 0, 1, 2, 3, 4 in left graph, we can color the graph using 3 graph, we can color the graph using 3 colors. But if we consider the vertices 0, 1, colors. But if we consider the vertices 0, 1, 2, 3, 4 in right graph, we need 4 colors2, 3, 4 in right graph, we need 4 colors

Analysis of Basic AlgorithmAnalysis of Basic Algorithm

WelshWelsh Powell Powell AlgorithmAlgorithm

Find the degree of each vertexFind the degree of each vertex ListList the verices the verices in order of descending in order of descending

valence i.e. valence i.e. degree(v(i))>=degree(v(i+1))degree(v(i))>=degree(v(i+1)) Colour Colour the first vertex in the listthe first vertex in the list Go down the sorted list and color every Go down the sorted list and color every

vertex not connected to the colored vertex not connected to the colored vertices above the same color then cross vertices above the same color then cross out all colored vertices in the list.out all colored vertices in the list.

Welsh Powell AlgorithmWelsh Powell Algorithm

Repeat the process on the uncolored Repeat the process on the uncolored vertices with a new color-always working vertices with a new color-always working in descending order of degree until all in descending order of degree until all vertices are colored.vertices are colored.

Complexity Complexity of above algorithm = of above algorithm = O(nO(n22))

Welsh Powell Algorithm: Welsh Powell Algorithm: ExampleExample

Welsh Powell Algorithm: Welsh Powell Algorithm: Example Example

Any QueriesAny Queries

Thank YouThank You

Graph colouring

Engineering

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