This module introduces random graphs and probabilistic methods, providing a foundation for understanding their applications in graph theory. Students will explore key concepts and definitions.
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This module introduces the fundamental concepts of vertex cover and independent sets in graphs. Students will learn the definitions and properties of these concepts, which are crucial in solving various graph-related problems.
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This module delves into matchings in graphs, focusing on König's theorem and Hall's theorem. Students will explore these important theorems, which provide key insights into the existence of matchings in bipartite graphs.
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This module continues the exploration of Hall's theorem, offering deeper insights and real-world applications. Students will analyze various scenarios where Hall's theorem is applicable, reinforcing their understanding of matchings in graphs.
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This module presents Tutte's theorem, which addresses the existence of perfect matchings in graphs. Students will study the theorem's statement, proof, and implications for graph theory and combinatorial optimization.
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This module builds on Tutte's theorem, providing further insights and examples to illustrate its applications. Students will explore various graph types and how Tutte's theorem can be applied to find perfect matchings.
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This module provides an in-depth look at various matching algorithms, exploring their complexities and applications. Students will learn about the theoretical foundations and practical implementations of these algorithms.
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This module focuses on dominating sets and path covers in graphs. Students will learn the definitions, properties, and significance of these concepts in graph theory.
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This module examines Gallai-Milgram theorem and Dilworth's theorem, crucial tools in combinatorial optimization. Students will learn the theorems' statements, proofs, and applications in various contexts.
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This module introduces the concept of connectivity in graphs, focusing on 2-connected and 3-connected graphs. Students will learn how to identify and analyze these types of graphs and their significance.
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This module covers Menger's theorem, which addresses connectivity in graphs. Students will explore the theorem's statement, proof, and implications for understanding graph connectivity.
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This module further explores connectivity concepts, focusing on k-linkedness in graphs. Students will learn about the definitions, properties, and significance of k-linkedness in graph theory.
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This module introduces minors and topological minors in graphs. Students will learn the definitions, properties, and significance of these concepts in the context of graph theory.
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This module focuses on vertex coloring, specifically Brooks' theorem. Students will learn about the theorem's statement, proof, and applications in graph theory.
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This module further explores vertex coloring, discussing advanced topics and techniques. Students will analyze various strategies for coloring graphs and their applications.
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This module introduces edge coloring, focusing on Vizing's theorem. Students will learn the theorem's statement, proof, and significance in graph theory.
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This module presents the proof of Vizing's theorem and introduces planarity concepts. Students will analyze the theorem's implications for edge coloring and explore the basics of planar graphs.
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This module discusses the 5-coloring of planar graphs and introduces Kuratowski's theorem. Students will learn about the theorem's statement, proof, and relevance in graph theory.
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This module provides the proof of Kuratowski's theorem and discusses list coloring. Students will analyze the theorem's implications and learn about the significance of list coloring in graph theory.
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This module focuses on the list chromatic index, exploring its definition, properties, and significance in graph theory. Students will learn about its applications in various contexts.
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This module introduces the adjacency polynomial of a graph and the combinatorial Nullstellensatz. Students will explore the definitions, properties, and applications of these concepts in graph theory.
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This module discusses the chromatic polynomial and k-critical graphs, providing insights into their definitions, properties, and applications in graph theory.
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This module examines the Gallai-Roy theorem, acyclic coloring, and Hadwiger's conjecture. Students will analyze the theorem's statement, proof, and implications in graph theory.
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This module presents examples of perfect graphs, illustrating their properties and applications. Students will learn how perfect graphs relate to other graph classes and their significance in graph theory.
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This module discusses interval graphs and chordal graphs, focusing on their definitions, properties, and applications in graph theory. Students will learn how these graphs relate to perfect graphs.
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This module presents the proof of the weak perfect graph theorem (WPGT), exploring its implications and significance in the study of perfect graphs. Students will analyze the theorem's proof and applications.
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This module provides a second proof of the weak perfect graph theorem and discusses some non-perfect graph classes. Students will explore the significance of these classes in the context of perfect graphs.
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This module explores more special classes of graphs, providing insights into their properties and significance in graph theory. Students will analyze various graph types and their applications.
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This module covers boxicity, sphericity, and Hamiltonian circuits, exploring their definitions, properties, and significance in graph theory. Students will analyze various applications in optimization.
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This module further explores Hamiltonicity, focusing on Chvátal's theorem. Students will learn about the theorem's statement, proof, and implications in graph theory.
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This module revisits Chvátal's theorem and discusses toughness, Hamiltonicity, and the 4-color conjecture. Students will analyze their significance and applications in graph theory.
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This module introduces network flows, focusing on the max flow mincut theorem. Students will learn about the theorem's statement, proof, and applications in optimization problems.
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This module expands on network flows, discussing circulations and their applications. Students will learn about the definitions, properties, and significance of circulations in network design.
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This module focuses on circulations and tensions in networks, providing insights into their definitions and applications. Students will explore how these concepts relate to network flows.
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This module further explores circulations and tensions, discussing flow numbers and Tutte's flow conjectures. Students will analyze the significance of these concepts in network flows.
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This module introduces random graphs and probabilistic methods, providing a foundation for understanding their applications in graph theory. Students will explore key concepts and definitions.
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This module focuses on the probabilistic method, discussing Markov's inequality and Ramsey numbers. Students will learn about the significance of these concepts in graph theory.
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This module delves into the probabilistic method, focusing on graphs of high girth and high chromatic number. Students will learn about their implications in graph theory.
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This module discusses the second moment method and Lovász local lemma, key concepts in the probabilistic method. Students will explore their significance and applications in graph theory.
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This module introduces graph minors and Hadwiger's conjecture, focusing on their definitions, properties, and significance in graph theory. Students will analyze implications in combinatorial optimization.
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This module focuses on more advanced topics related to graph minors, including tree decompositions. Students will learn about the significance and applications of these concepts in graph theory.
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