Discrete Mathematics - Vol 3 | 6. Question 9: Proving a Graphic Sequence by Abraham | Learn Smarter
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6. Question 9: Proving a Graphic Sequence

The chapter explores various aspects of graph theory, particularly focusing on graphic sequences, edge coloring, and vertex coloring. It discusses proofs and strategies for determining the chromatic number of complete graphs based on whether the number of vertices is odd or even. Additionally, the chapter presents counterexamples to illustrate limitations in greedy coloring strategies.

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Sections

  • 6

    Question 9: Proving A Graphic Sequence

    This section explores how to determine if a degree sequence is a graphic sequence using the Havel-Hakimi theorem or through constructive proofs.

    Learning Practice

  • 6.1

    Question 10: Edge Colouring In Graphs

    This section discusses the principles of edge colouring in graphs, specifically focusing on the conditions when a single colour can be used for a set of edges based on the parity of the number of vertices.

    Learning Practice

  • 6.2

    Question 11: Edge Chromatic Number Of Complete Graphs

    This section discusses determining the edge chromatic number of complete graphs, distinguishing between cases when the number of vertices is odd or even.

    Learning Practice

  • 6.2.1

    Case When N Is Even

    This section covers the concepts related to graphic sequences and edge coloring in graphs, specifically addressing scenarios when the number of vertices n is even.

    Learning Practice

  • 6.2.2

    Case When N Is Odd

    The section discusses the properties and implications of edge coloring in graphs with odd numbers of vertices and describes how the Havel-Hakimi theorem applies to prove a graphic sequence.

    Learning Practice

  • 6.3

    Question 12: Greedy Strategy For Vertex Colouring

    This section discusses the Welsh-Powell algorithm for vertex colouring, highlighting that it does not always yield the optimal solution.

    Learning Practice

References

ch55 - part B.pdf

Class Notes

Memorization

What we have learnt

  • Graphic sequences can be pr...
  • Edge coloring in graphs dep...
  • Vertex coloring strategies ...

Final Test

Revision Tests