29. Introduction to Tutorial 8
This chapter delves into fundamental concepts and properties of graphs, including Ramsey numbers, articulation points, trees, self-complementary graphs, and regular graphs. A strong focus is placed on proving or disapproving specific propositions regarding graph properties while providing a theoretical framework for understanding these relationships. Key proofs are reinforced through various exercises and activities, which encourage deeper engagement with the material.
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What we have learnt
- The complement of a graph has vertices that are the same as the original graph, with edges representing the absence of edges in the original graph.
- For any party with 6 guests, there will always exist either 3 mutual friends or 3 mutual enemies, demonstrating the properties of Ramsey numbers.
- A tree with n nodes possesses exactly n - 1 edges, an important property that supports numerous graph theory applications.
Key Concepts
- -- Ramsey Numbers
- A Ramsey number R(m, n) is the smallest number of vertices required to ensure that a graph contains a complete subgraph of m vertices or an independent set of n vertices.
- -- Articulation Point
- An articulation point (or cut vertex) in a graph is a vertex that, when removed along with its incident edges, increases the number of connected components of the graph.
- -- SelfComplementary Graph
- A graph is said to be self-complementary if it is isomorphic to its complement, meaning it can be transformed into its complement by a relabeling of vertices.
- -- Regular Graph
- A regular graph is a graph where each vertex has the same number of edges, known as the degree of the graph.
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