Ubiquity of graphs with nowhere-linear end structure

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.