Graph Theory

  1. Graph Coloring
    1. Vertex Coloring
      1. Problem Definition
        1. Proper Coloring
          1. Color Classes
          2. Chromatic Number
            1. Bounds and Estimates
              1. Computational Complexity
              2. Coloring Algorithms
                1. Greedy Coloring Algorithm
                  1. Vertex Ordering Strategies
                    1. Performance Analysis
                    2. Backtracking Approaches
                    3. Theoretical Results
                      1. Brooks' Theorem
                        1. Statement and Proof Outline
                          1. Exceptions and Applications
                          2. Four Color Theorem
                            1. Historical Significance
                              1. Computer-Assisted Proof
                                1. Planar Graph Coloring
                            2. Edge Coloring
                              1. Problem Definition
                                1. Proper Edge Coloring
                                  1. Color Classes for Edges
                                  2. Edge Chromatic Number
                                    1. Relationship to Maximum Degree
                                    2. Vizing's Theorem
                                      1. Class 1 and Class 2 Graphs
                                        1. Characterization Results
                                        2. Edge Coloring Algorithms
                                          1. Greedy Approaches
                                            1. Optimal Algorithms for Special Cases
                                          2. Applications of Graph Coloring
                                            1. Scheduling Problems
                                              1. Exam Scheduling
                                                1. Course Timetabling
                                                  1. Resource Allocation
                                                  2. Register Allocation
                                                    1. Compiler Optimization
                                                      1. Interference Graphs
                                                      2. Frequency Assignment
                                                        1. Wireless Networks
                                                          1. Radio Broadcasting
                                                          2. Map Coloring
                                                            1. Geographic Applications
                                                          3. Chromatic Polynomials
                                                            1. Definition
                                                              1. Polynomial Function
                                                                1. Deletion-Contraction Recurrence
                                                                2. Properties
                                                                  1. Degree and Leading Coefficient
                                                                    1. Roots and Their Significance
                                                                    2. Calculation Methods
                                                                      1. Recursive Computation
                                                                        1. Special Graph Families

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                                                                    10. Planar Graphs