[Paper Review] Linear Connectivity Forces Large Complete Bipartite Minors: the Patch for the Large Tree-Width Case

Jan-Oliver Fröhlich, Theodor Müller|ArXiv.org|Jun 15, 2009

Advanced Graph Theory Research4 references3 citations

TL;DR

This paper provides a rigorous patch to a gap in Böhme et al.'s proof that linearly high connectivity forces large complete bipartite minors, specifically in graphs of large tree-width. By replacing their unproven structure theorem with a direct application of Robertson and Seymour’s original structure theorem, the authors establish the existence of a large, wide vortex in the embedded core graph, ensuring the necessary comb structure for the minor construction, thus validating the main result under the original assumptions.

ABSTRACT

The recent paper "Linear Connectivity Forces Large Complete Bipartite Minors" by Boehme et al. relies on a structure theorem for graphs with no H-minor. The sketch provided of how to deduce this theorem from the work of Robertson and Seymour appears to be incomplete. To fill this gap, we modify the main proof of that paper to work with a mere restatement of Robertson and Seymour's original results instead.

Motivation & Objective

• To address a critical gap in the proof of Theorem 1.1 from Böhme et al.'s paper on complete bipartite minors.
• To replace the unproven structure theorem (Theorem 4.2) in their argument with a direct, rigorous application of Robertson and Seymour’s original structure theorem.
• To ensure the existence of a large, wide vortex in the embedded core graph, which supports the minor construction.
• To validate the main result of Böhme et al. by providing a self-contained, correct argument that works within the original framework.
• To define essential vertices based on low degree in the core graph and use this to prove the existence of an $ n_2 $-wide vortex.

Proposed method

• Restate Robertson and Seymour’s structure theorem in terminology consistent with Böhme et al.’s paper for direct applicability.
• Define a vortex as a graph with a linearly ordered society and a linked path-decomposition with controlled adhesion sets.
• Use the concept of $ \alpha $-near-embedding into a surface, with apex sets, large vortices (linked, high adhesion), and small vortices (length ≤ 3).
• Introduce the notion of essential society vertices (degree < 7 in $ G_0 $) to refine the Euler characteristic argument.
• Apply the Euler formula to the embedded core graph $ G_0 $, using bounds on edges and faces to derive a lower bound on the number of essential vertices.
• Use Corollary 5 to bound the number of branch sets of a large wall minor intersecting vortices and apex sets, leading to a contradiction if $ G_0 $ is too small.

Experimental results

Research questions

• RQ1 Does the structure theorem used by Böhme et al. to deduce the existence of a large embedded wall in high tree-width graphs hold rigorously?
• RQ2 Can the existence of a large, wide vortex in the embedded core graph be established without relying on their unproven Theorem 4.2?
• RQ3 Is the Euler characteristic argument valid when using the revised definition of essential vertices?
• RQ4 Can the original proof of Theorem 1.1 be repaired using only Robertson and Seymour’s foundational results?
• RQ5 Does the presence of a comb disjoint from the linkage, with teeth at the society vertices, follow from Theorem 1 in the revised framework?

Key findings

• The core graph $ G_0 $ must contain more than $ n_1 $ vertices, as otherwise the number of branch sets of a large wall minor would be insufficient to match the required $ r^2 $ count.
• At most $ g $ small vortices exist, as each is separated by at most 3 vertices and contributes few branch sets to the wall minor.
• The number of essential society vertices in $ G_0 $ exceeds $ \frac{1}{7}|G_0| - 2(a+1)g - ask $, ensuring a large number of such vertices.
• An $ n_2 $-wide vortex exists, meaning its society contains at least $ n_2 $ essential vertices, which is sufficient to trigger the minor construction.
• The existence of a comb disjoint from the linkage of a large vortex is guaranteed by Theorem 1, ensuring the minor construction proceeds correctly.
• The revised definition of essential vertices and the Euler formula argument now hold, validating the critical step in the original proof.