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The χ -Binding Function of d-Directional Segment Graphs.

Lech Duraj1, Ross J Kang2, Hoang La1

  • 1Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Krakow, Poland.

Discrete & Computational Geometry
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Summary
This summary is machine-generated.

This study explores intersection graphs of line segments with limited slopes (d-DIR). Researchers construct graphs that precisely meet the theoretical upper bound for chromatic number (χ) when the clique number (ω) is even, confirming a conjecture.

Keywords:
Chi-boundednessGeometric graphsGraph colouringSegment graphs

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Area of Science:

  • Graph theory
  • Computational geometry
  • Combinatorics

Background:

  • Intersection graphs of line segments are fundamental objects in geometric graph theory.
  • The class d-DIR comprises intersection graphs of line segments in R² with at most d slopes.
  • Previous work established bounds on the chromatic number (χ) based on the clique number (ω) for specific cases.

Purpose of the Study:

  • To investigate the exact relationship between the chromatic number and clique number for graphs in the d-DIR class.
  • To construct graphs within d-DIR that achieve the maximum possible chromatic number for a given clique number.
  • To determine the precise χ-binding function for the d-DIR class.

Main Methods:

  • Graph construction techniques were employed to create specific instances of d-DIR graphs.
  • Theoretical analysis was used to establish bounds and confirm the tightness of these bounds.
  • The study extends existing results by considering a general number of slopes, d.

Main Results:

  • For graphs in d-DIR, the chromatic number (χ) is bounded by dω, where ω is the clique number.
  • Graphs were constructed that achieve this dω bound exactly when ω is even, partially confirming a conjecture.
  • The study determined the exact χ-binding function for d-DIR: dω for even ω and d(ω-1)+1 for odd ω.

Conclusions:

  • The χ-binding function for d-DIR graphs is fully characterized.
  • The findings provide a precise understanding of the chromatic number's behavior in relation to the clique number for this class of geometric graphs.
  • This work generalizes and extends previous results on interval graphs and related structures.