Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Discrete Mathematics → Graph Theory ↓


Planarity and Coloring


Graph theory is an important field within discrete mathematics, dealing with graphs which are structures used to model paired relationships between objects. In this vast field, two essential concepts are flatness and graph coloring. Understanding these concepts provides insight into how we can view and color graphs in a way that satisfies required mathematical properties.

Flatness in graphs

A graph is planar if it can be drawn on a plane without any of its edges crossing each other. Planarity is important because it determines whether a graph can be represented in two-dimensional space without any intersections, which is particularly useful in circuit design, cartography, and more.

To determine whether a graph is planar, we use the Kuratowski theorem, which states:

A finite graph is planar if and only if it does not contain any subgraph that is a subpartition of the complete graph K 5 (a complete graph on five vertices) or the complete bipartite graph K 3,3 (a bipartite graph with two sets of three vertices).

K5 and K3.3

K 5 graph and K 3,3 graph are important in understanding graph flatness:

K 5 : A graph in which 5 vertices are connected to each other, resulting in 10 edges.


    
    
    
    
    
    
    
    
    
    
    
    
    
    
    



K 3,3 : a bipartite graph with two sets of three vertices, such that every vertex of one set is connected to all vertices of the other set.

Flatness test

To determine if a graph is planar, we need to check for a subgraph congruent to K 5 or K 3,3. If either of these exists, the graph is non-planar. The most common algorithm used to check planarity is the planarity testing algorithm, which uses recursive partitioning of the graph to find subdivisions of a non-planar graph.

Try drawing a graph without crossings and use subgraph checking for graphs with less than 10 vertices:

1. Try to display the graph on a plane.
2. If this is difficult, use the Kuratowski theorem to find out whether any subdivisions of K 5 or K 3,3 exist.
3. If a subdivision is found, then the graph is non-planar.

Graph coloring

Graph coloring involves assigning colors to graph elements under specific restrictions. The most common is vertex coloring, which aims to color the vertices such that no two adjacent vertices share the same color. The minimum number of colors required for such a coloring of a graph is the chromatic number of the graph, usually denoted as χ(G).

Basic principles of coloring

The simplest form of graph coloring is a proper coloring, which uses only the basics of ensuring that adjacent vertices have distinct colors. An essential theorem related to graph coloring is the four color theorem, which states that any planar graph can be colored using at most four colors.

Examples of graph coloring

Example 1: Color the cycle graph C 4

Here, we use only two colors (red and blue) to ensure that no two adjacent vertices share the same color.

Example 2: Color a complete graph K 3

In a complete graph K n, every vertex is connected to every other vertex. Therefore, we must use n different colors. In this case, K 3 requires three different colors: red, blue, and green.

Applications of graph coloring

Graph coloring has a variety of applications, extending beyond theoretical exercises to practical problems, such as:

  • Allocation of resources (for example, register allocation in the compiler)
  • Scheduling issues (ensuring that no two overlapping tasks share the same resources)
  • Frequency determination problems (assigning frequencies to transmitters in a manner that minimizes overlapping frequencies)

Through these applications, the widespread utility and practicality of graph coloring in optimizing the efficient use of resources becomes apparent.

Conclusion

Planarity and coloring are fundamental concepts in graph theory, which significantly impact mathematical theory and practical applications. Understanding which graphs can be embedded in the plane without crossings and how they can be colored to satisfy constraints provides a fundamental framework for solving complex problems in a variety of domains. Mastering these concepts allows mathematicians and computer scientists to handle complex network and relational models with confidence and efficiency.


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