# Graph Theory Topics

## 1. Types and Graphs and Basic Definitions

- Graphs and Subgraphs
- Adjacency and Incidence
- Neighbourhood, Degrees and Degree Sequence
- The Maximum and Minimum Degrees of a Graph
- Isolated Vertices, Leaves, and Pendant Edges
- Realizability of Graphs
- Graph Isomorphisms
- Walks, Trails, Paths, Cycles, and Circuits
- Graph Bridges
- Components of a Graph
- The Handshaking Lemma
- Regular Graphs
- Complete Graphs
- Cycle Graphs
- The Circumference and Girth of a Graph
- Null Graphs
- Platonic Graphs
- Bipartite and Complete Bipartite Graphs
- Tree Graphs
- Path Graphs
- Cube Graphs
- Complement Graphs
- Self Complementary Graphs
- The Instant Insanity Problem

## 2. Eulerian and Hamiltonian Graphs

- Eulerian Graphs and Semi-Eulerian Graphs
- Euler's Theorem
- Hamiltonian Graphs and Semi-Hamiltonian Graphs
- Dirac's and Ore's Theorem
- Knight's Tour Problem
- Gray Codes
- Digraphs
- Subdigraphs and Underlying Graphs
- Disconnected, Connected, and Strongly Connected Digraphs
- Orientable Graphs
- Out-Degree Sequence and In-Degree Sequence
- The Handshaking Dilemma
- Eulerian Digraphs
- Hamiltonian Digraphs
- The Rotating Drum Problem
- Tournaments and Rankings
- Tournament Kings
- Adjacency Matrices
- Incidence Matrices

## 3. Tree Graphs

## 4. Vertex and Edge Cutsets and Euler's Formula

- Vertex and Edge Cutsets
- Relation of Min Vertex Cutsets, Min Edge Cutsets, The Minimum Degree, and Maximum Degree
- Euler's Formula
- Planar Graphs and Plane Drawings
- Proof of the Existence of only 5 Platonic Solids
- Conditions for Planarity
- Algebraic Proof of the Existence of only 5 Platonic Solids
- Subdivisions and Contractions
- Kuratowski's Theorem
- Graph Duality

## 5. Vertex and Edge Colourings

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###### References

- 1. "Graphs and Applications: An Introductory Approach" by Joan M. Aldous and Robin J. Wilson.