[Paper Review] Ubiquity in graphs II: Ubiquity of graphs with non-linear end structure

Nathan Bowler, Christian Elbracht+5 more|arXiv (Cornell University)|Sep 3, 2018

Advanced Graph Theory Research10 references3 citations

TL;DR

This paper establishes a sufficient condition on the end structure of locally finite graphs to guarantee $/preceq$-ubiquity under the minor relation, proving that graphs with non-linear end structures—such as the full grid—are $/preceq$-ubiquitous. The result supports Andreae's conjecture by identifying structural properties that ensure ubiquity.

ABSTRACT

A graph $G$ is said to be $\preceq$-ubiquitous, where $\preceq$ is the minor relation between graphs, if whenever $\Gamma$ is a graph with $nG \preceq \Gamma$ for all $n \in \mathbb{N}$, then one also has $\aleph_0 G \preceq \Gamma$, where $\alpha G$ is the disjoint union of $\alpha$ many copies of $G$. A well-known conjecture of Andreae is that every locally finite connected graph is $\preceq$-ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph~$G$ which implies that $G$ is $\preceq$-ubiquitous. In particular this implies that the full grid is $\preceq$-ubiquitous.

Motivation & Objective

• To investigate the ubiquity of locally finite connected graphs under the minor relation.
• To identify structural conditions on the ends of a graph that ensure $/preceq$-ubiquity.
• To provide a sufficient condition for $/preceq$-ubiquity that applies to graphs with non-linear end structures.
• To contribute to the resolution of Andreae's conjecture on the ubiquity of locally finite graphs.

Proposed method

• The authors analyze the topological structure of ends in locally finite graphs, focusing on non-linear end structures.
• They define a condition on the end structure that ensures the existence of arbitrarily large finite minors in any graph $Γ$ containing $nG$ for all $n \in \mathbb{N}$.
• The method relies on combinatorial and topological arguments concerning the connectivity and separation properties of ends.
• The proof uses the concept of minor-closed families and applies known results on infinite graphs and their minors.
• It establishes that if the end structure is non-linear, then the graph satisfies the ubiquity condition.
• The argument culminates in showing that the full grid, which has a non-linear end structure, is $/preceq$-ubiquitous.

Experimental results

Research questions

• RQ1 Under what structural conditions on the ends of a locally finite graph is the graph $/preceq$-ubiquitous?
• RQ2 Does the presence of non-linear end structures in a graph imply $/preceq$-ubiquity under the minor relation?
• RQ3 Can the full grid be proven $/preceq$-ubiquitous using end structure conditions?
• RQ4 Does the proposed condition on end structure suffice to verify Andreae's conjecture for a broad class of graphs?
• RQ5 How do non-linear end structures influence the existence of infinite minors in graphs containing arbitrarily large finite disjoint unions of a graph G?

Key findings

• A sufficient condition on the end structure of a locally finite graph ensures that it is $/preceq$-ubiquitous under the minor relation.
• Graphs with non-linear end structures satisfy the ubiquity condition, meaning that if $nG \preceq \Gamma$ for all $n$, then $\aleph_0 G \preceq \Gamma$.
• The full grid is proven to be $/preceq$-ubiquitous as a direct consequence of its non-linear end structure.
• The result provides a significant step toward resolving Andreae's conjecture by identifying a broad class of graphs for which ubiquity holds.
• The method establishes a structural criterion—non-linearity of ends—that guarantees ubiquity, independent of other graph properties.
• The findings demonstrate that end structure alone can determine the ubiquity of a graph under the minor relation.