3. Vertex and Edge Colouring
The lecture focuses on vertex and edge colouring in graph theory, emphasizing their applications such as exam scheduling and tournament match planning. It explains the concepts of vertex chromatic number and edge chromatic number, along with greedy algorithms for colouring. The complexities and challenges associated with finding the chromatic numbers are highlighted, alongside upper and lower bounds on these values.
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Sections
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What we have learnt
- Vertex colouring ensures no two adjacent vertices in a graph share the same colour.
- The vertex chromatic number is the minimum number of colours required for a proper vertex colouring.
- The edge chromatic number is the minimum number of colours needed to colour the edges such that no two adjacent edges share the same colour.
Key Concepts
- -- Vertex Colouring
- A method of assigning colours to the vertices of a graph so that no two adjacent vertices have the same colour.
- -- Vertex Chromatic Number (χ(G))
- The minimum number of colours needed to colour the vertices of a graph without adjacent vertices sharing the same colour.
- -- Edge Colouring
- A method of assigning colours to the edges of a graph such that no two edges that share a vertex have the same colour.
- -- Edge Chromatic Number (χ₀(G))
- The minimum number of colours required to colour the edges of a graph without adjacent edges sharing the same colour.
- -- Greedy Algorithm for Colouring
- An approach to colouring where each vertex is coloured with the first available colour not assigned to its adjacent vertices.
- -- GuptaVizing Theorem
- A theorem establishing that the edge chromatic number of a simple graph is bounded by the maximum degree of that graph plus one.
Additional Learning Materials
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