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Graph Coloring

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Basic Graph Theory

Part of the book series: Undergraduate Topics in Computer Science ((UTICS))

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Abstract

Probably graph coloring concept naturally arose from its application in map coloring: given a map containing several countries, we wish to color the countries in the map in such a way that neighboring countries receive different colors to make the countries distinct. In this chapter we know about vertex coloring, edge coloring, chromatic number, chromatic index, chromatic polynomial, etc.

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Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, Bangladesh University of Engineering and Technology (BUET), Dhaka, Bangladesh

    Md. Saidur Rahman

Authors
  1. Md. Saidur Rahman
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Corresponding author

Correspondence to Md. Saidur Rahman .

Exercises

Exercises

Fig. 7.9
figure 9

Graphs

Full size image
  1. 1.

    Obtain vertex coloring of the graphs in Fig. 7.9 with the minimum number of colors.

  2. 2.

    Determine the chromatic number of Petersen graph.

  3. 3.

    Show that dual graph of a maximal plane graph is a cubic graph.

  4. 4.

    Let G be a simple planar graph containing no triangle. Then show that \(\chi (H)\le 4\).

  5. 5.

    Show that for a tree of n vertices \(P_G(k)=k(k-1)^{n-1}\).

  6. 6.

    Construct a 4-critical graph.

  7. 7.

    Let G be a k-critical graph. Then show that the degree of every vertex of G is at least \(k-1\).

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Rahman, M.S. (2017). Graph Coloring. In: Basic Graph Theory. Undergraduate Topics in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-319-49475-3_7

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  • DOI: https://doi.org/10.1007/978-3-319-49475-3_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-49474-6

  • Online ISBN: 978-3-319-49475-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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