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Ubiquity of graphs with nowhere-linear end structure.

Nathan Bowler1, Christian Elbracht1, Joshua Erde2

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Graphs are -ubiquitous if they contain specific graph minors. This study provides a structural condition on graph ends that guarantees -ubiquity, proving the full-grid is -ubiquitous.

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graph minorsinfinite graphsubiquity

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Area of Science:

  • Graph Theory
  • Combinatorics
  • Topology

Background:

  • A graph is -ubiquitous if it contains specific graph minors.
  • Andreae's conjecture states all locally finite connected graphs are -ubiquitous.
  • Understanding graph ubiquity is crucial for structural graph theory.

Purpose of the Study:

  • To provide a sufficient condition for a graph to be -ubiquitous.
  • To explore the relationship between graph ends and -ubiquity.
  • To confirm the -ubiquity of the full-grid graph.

Main Methods:

  • Analysis of the structure of graph ends.
  • Development of a sufficient condition based on end structure.
  • Application of the condition to specific graph families.

Main Results:

  • A novel sufficient condition for -ubiquity is established.
  • The condition relates to the structure of a graph's ends.
  • The full-grid graph is proven to be -ubiquitous.

Conclusions:

  • The findings offer new insights into graph ubiquity.
  • The results contribute to verifying Andreae's conjecture.
  • The study advances the understanding of graph structure and minors.