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Coloring triangle-free graphs with local list sizes.

Ewan Davies1, Rémi de Joannis de Verclos2, Ross J Kang2

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Summary
This summary is machine-generated.

This study refines results on the list chromatic number of triangle-free graphs by allowing lower color bounds for low-degree vertices. Findings impact list, correspondence, and fractional coloring theories.

Keywords:
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Area of Science:

  • Graph Theory
  • Combinatorics
  • Discrete Mathematics

Background:

  • Recent breakthroughs by Molloy and Bernshteyn advanced understanding of the list chromatic number in triangle-free graphs.
  • The list chromatic number is a key concept in graph coloring, measuring the minimum number of colors needed for a proper coloring from given lists of colors at each vertex.

Purpose of the Study:

  • To provide two distinct and natural refinements of existing results on the list chromatic number of triangle-free graphs.
  • To explore relaxations in color availability for vertices with lower degrees.
  • To investigate implications for list coloring, correspondence coloring, and fractional coloring.

Main Methods:

  • Development of two novel refinement techniques for list coloring bounds.
  • Application of the hard-core model to prove a Johansson-type result in fractional coloring.
  • Analysis of color availability adjustments based on vertex degree.

Main Results:

  • Two new refinements of the list chromatic number for triangle-free graphs are established.
  • A result concerning list coloring and correspondence coloring is presented.
  • A new result in fractional coloring is demonstrated, utilizing the hard-core model.

Conclusions:

  • The study offers more nuanced bounds for coloring triangle-free graphs, particularly for vertices with lower degrees.
  • The techniques developed, especially the use of the hard-core model, may have broader applications in graph theory and related fields.