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On 3-Coloring of ( )-Free Graphs.

Vít Jelínek1, Tereza Klimošová1, Tomáš Masařík1,2,3

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This study solves the 3-coloring problem for a specific type of graph called P8-free graphs. This finding advances the understanding of graph coloring complexity for hereditary graph classes.

Keywords:
3-ColoringCographsHereditary classes

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Area of Science:

  • Graph Theory
  • Computational Complexity

Background:

  • The 3-coloring problem for hereditary graph classes is a significant area of research.
  • Complexity is known for H-free graphs up to seven vertices, with two unsolved cases on eight vertices.

Purpose of the Study:

  • To investigate the complexity of the 3-coloring problem for P8-free graphs.
  • To determine if 3-coloring is efficiently solvable for this previously unexplored class.

Main Methods:

  • The study focuses on P8-free graphs, a class defined by forbidden induced subgraphs.
  • Algorithmic techniques are employed to develop a polynomial-time solution.

Main Results:

  • The 3-coloring problem is proven to be polynomial-time solvable for P8-free graphs.
  • This resolves one of the remaining open cases in the complexity of 3-coloring for H-free graphs.

Conclusions:

  • The 3-coloring problem is efficiently solvable on P8-free graphs.
  • This contributes to a more complete understanding of graph coloring complexity in hereditary graph classes.