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

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.