Small counterexamples to the fat minor conjecture

TL;DR

This work advances the study of $K$-fat minors in coarse graph theory by producing significantly smaller incompressible graphs than previously known, including $K_t$ for $t\ge 6$, $K_{s,t}$ for $s,t\ge 4$, and $K_{2,2,2}$, via NSS-graph-based constructions. It introduces the coarse self-similarity of the NSS family and leverages a 2-fat model framework to propagate incompressibility through quasi-isometries, ultimately proving strong non-fat-minor results (e.g., $O$ not a $9$-fat minor) and deriving that certain complete and complete bipartite graphs are incompressible. The paper also shows that suspending counterexamples preserves incompressibility, enabling a cascade of new incompressible graphs (e.g., higher $K_t$ and $K_{s,t}$ families) and establishes asymptotic existence of $K_{3,t}$ and $K_5$ fat minors within NSS graphs. These results sharpen the boundary between graphs that satisfy and fail the fat-minor conjecture, and frame several open problems about the uniqueness of the OSS-type counterexamples and the limits of NSS-based methods in coarse Kuratowski-type questions.

Abstract

We narrow the gap between the family of graphs that do and the family of graphs that do not satisfy the fat minor conjecture by obtaining much simpler counterexamples than were previously known, including $K_t, t \geq 6$ and $K_{s,t}, s,t \geq 4$ and $K_{2,2,2}$. This is achieved by establishing a `coarse self-similarity' property of the graphs used by Nguyen, Scott and Seymour to disprove the `coarse Menger conjecture'. This property may be of independent interest.

Figures (17)

• Figure 1: Depicted is the graph $G_{5,d}$. The dotted curves represent paths of length $d+1$.
• Figure 2: The blue subgraph of $G := G_{4,d}$ is the trigon $\mathop{\mathrm{S\Delta}}\nolimits(G)$.
• Figure 3: A nested linkage (left) and a crossing linkage (right).
• Figure 4: Indicated in blue is the unique path in $B(G)$ between $\mathop{\mathrm{S_1}}\nolimits(\Delta)$ and $\mathop{\mathrm{T_1}}\nolimits(\Delta)$, where $\Delta$ is indicated in black.
• Figure 5: Depicted are the branch sets of a model of a $K_7-\{16,26\}$ in $G_{7,d}$. The dark gray area in the bottom picture represents the graph from the top picture.

Theorems & Definitions (58)

• Conjecture 1.1: GeoPapMin
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Proposition 1.7
• Lemma 2.1
• Lemma 2.2