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Single-conflict colouring.

Zdeněk Dvořák1, Louis Esperet2, Ross J Kang3

  • 1Computer Science Institute (CSI) Charles University Prague Czech Republic.

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Summary
This summary is machine-generated.

Researchers explored the single-conflict chromatic number for graphs on surfaces. They found this number is bounded by a function of the surface

Keywords:
DP‐colouringadaptable choosabilitygraphs on surfaceslist colouringsingle‐conflict chromatic number

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Area of Science:

  • Graph Theory
  • Combinatorics
  • Topological Graph Theory

Background:

  • Introduces the concept of local color assignments to edges incident to vertices in a multigraph.
  • Defines the single-conflict chromatic number as the minimum number of colors needed to ensure a conflict-free selection of one color per vertex.
  • Highlights the relationship of this parameter to separation choosability and adaptable choosability.

Purpose of the Study:

  • To determine the single-conflict chromatic number for simple graphs embeddable on surfaces with a given Euler genus.
  • To establish bounds for this chromatic number based on the topological properties of the graph's embedding surface.

Main Methods:

  • Utilizes concepts from graph theory and combinatorics.
  • Applies techniques related to graph embeddings and choosability parameters.
  • Develops theoretical arguments to establish bounds and prove results.

Main Results:

  • The single-conflict chromatic number for simple graphs embeddable on a surface of Euler genus g is shown to be O(sqrt(g)).
  • This bound is demonstrated to be sharp up to a logarithmic factor, indicating its tightness.

Conclusions:

  • The study provides a precise characterization of the single-conflict chromatic number for a significant class of graphs.
  • The findings contribute to the understanding of choosability parameters in topological graph theory.