10.2 Vertex coloring and chromatic number

Vertex coloring assigns colors to graph vertices, ensuring adjacent ones differ. It's crucial for solving real-world problems like map coloring and scheduling. The chromatic number represents the minimum colors needed for proper coloring.

Determining the chromatic number involves algorithms and theorems. Applications of vertex coloring are diverse, from solving Sudoku puzzles to creating exam timetables. The Four Color Theorem for planar graphs is a significant result in this field.

Vertex Coloring Fundamentals

Concept of vertex coloring

Top images from around the web for Concept of vertex coloring

• 

  Teoría de grafos - Colorear View original

  Is this image relevant?

• 

  VertexColoring | Wolfram Function Repository View original

  Is this image relevant?

• 

  Graph coloring - Wikipedia View original

  Is this image relevant?

• 

  Teoría de grafos - Colorear View original

  Is this image relevant?

• 

  VertexColoring | Wolfram Function Repository View original

  Is this image relevant?

1 of 3

Top images from around the web for Concept of vertex coloring

• 

  Teoría de grafos - Colorear View original

  Is this image relevant?

• 

  VertexColoring | Wolfram Function Repository View original

  Is this image relevant?

• 

  Graph coloring - Wikipedia View original

  Is this image relevant?

• 

  Teoría de grafos - Colorear View original

  Is this image relevant?

• 

  VertexColoring | Wolfram Function Repository View original

  Is this image relevant?

1 of 3

• Vertex coloring assigns colors to graph vertices ensuring adjacent vertices have different colors
• Proper vertex coloring uses minimum colors needed while maintaining color difference between adjacent vertices
• Vertex coloring solves real-world problems (map coloring, scheduling, frequency assignment in radio networks, register allocation in compiler optimization)

Determining chromatic number

• Chromatic number represents minimum colors required for proper vertex coloring
• Greedy coloring algorithm and backtracking algorithm help find chromatic number
• Upper bound: Brooks' theorem states $\chi(G) \leq \Delta(G) + 1$
• Lower bound: Clique number $\chi(G) \geq \omega(G)$
• Special cases include bipartite graphs (chromatic number 2), complete graphs (chromatic number $n$), cycle graphs (chromatic number 2 for even cycles, 3 for odd cycles)

Advanced Concepts and Applications

Planar graphs vs vertex coloring

• Four Color Theorem proves every planar graph is 4-colorable historically significant complex proof
• Face coloring in planar graphs dually relates to vertex coloring
• Outerplanar graphs possess 3-colorable property
• Graph thickness correlates with chromatic number

Applications of coloring algorithms

• Graph coloring heuristics employ largest degree first and saturation degree ordering techniques
• Vertex coloring algorithms solve Sudoku puzzles, create exam timetables, and schedule sports events
• Optimization uses integer programming formulation and local search algorithms
• Graph coloring is NP-complete requiring approximation algorithms for large graphs