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Essential Graph Theory: Key Concepts and Proofs for Chapter 10 Review

This comprehensive review covers essential topics in graph theory, focusing on basic concepts and proofs necessary for understanding Chapter 10. Key subjects include special graphs such as Kn, Cn, Wn, Qn, and Km,n, with an emphasis on their applications in calculations and counterexamples. The review details definitions of bipartite, isomorphic, and planar graphs, along with theorems for testing Euler circuits and paths. Prepare for unit tests by mastering calculations related to degrees, adjacency, paths, and chromatic numbers in planar graphs.

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Essential Graph Theory: Key Concepts and Proofs for Chapter 10 Review

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  1. Exam 3 review Chapter 10- Graphs

  2. Basics and proofs to know well • Special graphs: Kn, Cn, Wn, Qn, Km,n • Use these for calculations, counterexamples… • Know the definitions of bipartite, isomorphic, and planar • Prove: • Bipartite • or not • Isomorphic • or not • Planar • or not • Euler circuit or path possible • or not • Other proofs– be able to supply some details, as we do in class on harder problems

  3. Theorems to know for the unit test • 10.2: Thm. 1 Handshaking: 2e= sum of deg(v) • Thm. 2: undirected graph has an even # of odd degree • Conditions for when an Euler path or circuit exist (don’t worry about Hamilton conditions) • 10.7: Euler: r=e-v+2 • Cor 1: connected, planar, simple, e≤ ev-6 • Cor3: no circuits length 3, then e≤2v-4 • Thm. 2: A graph is nonplanariff it contains a subgraphhomeomorphic to K3,3or K5. • 10.8: Thm 1- chromatic # of planar graph ≤4

  4. Calculations to do • Calculate deg, deg-, deg+ • Adjacency tables and matrices • Paths • Strong and weakly connected • Counting paths of a certain length l • Euler and Hamilton paths and circuits • Conditions for Euler paths and circuits (not for Hamilton) • Chromatic number of special graphs

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