[Paper Review] Forcing large complete minors in infinite graphs

Maya Stein, José Zamora|arXiv (Cornell University)|Feb 2, 2011

Advanced Graph Theory Research5 references3 citations

TL;DR

This paper extends classical results on forcing large complete minors in finite graphs to infinite graphs by introducing a generalized notion of relative end degree. By combining minimum vertex degree with minimum relative end degree, the authors prove that infinite graphs with countably many ends and sufficiently high combined degree conditions contain large complete topological minors, generalizing Mader's and Kostochka-Bollobás-Thomason theorems to the infinite setting.

ABSTRACT

It is well-known that in finite graphs, large complete minors/topological minors can be forced by assuming a large average degree. Our aim is to extend this fact to infinite graphs. For this, we generalise the notion of the relative end degree, which had been previously introduced by the first author for locally finite graphs, and show that large minimum relative degree at the ends and large minimum degree at the vertices imply the existence of large complete (topological) minors in infinite graphs with countably many ends.

Motivation & Objective

• To extend the classical result that high average degree forces large complete minors in finite graphs to infinite graphs.
• To address the failure of minimum vertex degree alone to force large minors in infinite graphs due to infinite trees.
• To introduce and formalize the concept of relative end degree as a measure of density at ends of infinite graphs.
• To prove that a combination of high minimum vertex degree and high minimum relative end degree forces large complete topological minors in infinite graphs with countably many ends.
• To generalize Mader's and Kostochka-Bollobás-Thomason theorems to the infinite setting using a degree condition over vertices and ends.

Proposed method

• Generalize the notion of relative end degree from locally finite graphs to arbitrary infinite graphs with countably many ends.
• Define the relative degree of an end ω as the infimum over liminf of |∂eHi| / |∂vHi| for sequences of ω-regions (Hi) converging to ω.
• Introduce the combined minimum degree δV,Ω(G) as the minimum over all vertex degrees and relative end degrees in G.
• Use a recursive construction of finite subgraphs Si to build a sequence of sets with increasing average degree.
• Apply Kőnig’s infinity lemma to derive a contradiction from the existence of an infinite ray in a subgraph with bounded average degree.
• Leverage the structure of disjoint finite sets F ∈ F and their separation properties to control connectivity and degree growth.

Experimental results

Research questions

• RQ1 Can the classical result that high average degree forces large complete minors in finite graphs be extended to infinite graphs?
• RQ2 What additional structural condition is needed beyond high minimum vertex degree to force large complete minors in infinite graphs?
• RQ3 How can the concept of end degree be formalized in a way that captures density escaping to infinity in infinite graphs?
• RQ4 Is there a degree condition combining vertex and end degrees that guarantees the existence of large complete topological minors in infinite graphs with countably many ends?
• RQ5 To what extent does the relative end degree serve as a suitable replacement for average degree in infinite graph minor theorems?

Key findings

• The paper establishes that if the combined minimum degree δV,Ω(G) exceeds c1k√log k, then the complete graph Kk is a minor of G.
• For topological minors, if δV,Ω(G) > c2k², then Kk is a topological minor of G, extending the finite case result to infinite graphs.
• The existence of a ray with infinitely many vertices in a subgraph G′ contradicts the construction, implying that the process must terminate in a finite subgraph with high average degree.
• The construction of sets Si ensures that the average degree of vertices in G′[Si] exceeds m−k+1, leading to a finite subgraph with sufficient density to contain a large complete minor.
• The relative end degree provides a robust measure of density at ends, preventing the escape of density to infinity in infinite graphs.
• The proof relies on a recursive construction of finite subgraphs and Kőnig’s infinity lemma to derive a contradiction from the assumption that no large minor exists.