Verifying Hadwiger's Conjecture for Examples of Graphs with $α(G) = 2$

TL;DR

The paper tackles Hadwiger's Conjecture in the challenging α(G)=2 case by developing tools that transfer minor/chromatic properties through inflations. It systematically surveys counterexample structure, introduces inflation-based reasoning, and connects fractional clique coverings with inflated graphs to derive HC variants for broad graph classes. The authors prove Hadwiger-type results for inflations of complements of graphs with girth≥5, triangle-free Kneser graphs, and several strongly regular triangle-free graphs (notably Clebsch, Mesner, and Gewirtz; with Higman-Sims providing SHC_half), and extend these insights to Eberhard graphs. The work also clarifies the landscape of open questions and outlines a framework for identifying potential counterexamples via unavoidable graphs and Cayley constructions, highlighting the depth and richness of α(G)=2 Hadwiger-type problems.

Abstract

Hadwiger's Conjecture states that every graph with chromatic number $k$ contains a complete graph on $k$ vertices as a minor. This conjecture is a tremendous strengthening of the Four-Colour Theorem and is regarded as one of the most important open problems in graph theory. The case of Hadwiger's Conjecture for graphs with $α(G) = 2$ has garnered much attention. Seymour writes: ``My own belief is, if Hadwiger's Conjecture is true for graphs with stability number two then it is probably true in general, so it would be very nice to decide this case.'' This paper presents several tools useful for proving that a graph $G$ with $α(G) = 2$ satisfies Hadwiger's Conjecture. In doing so, we survey and generalise several classical results on the $α(G) = 2$ case of Hadwiger's Conjecture. Further, we apply these tools to prove variants of Hadwiger's Conjecture for several noteworthy classes of graphs with $α(G) = 2$. In particular, we prove Hadwiger's Conjecture for inflations of the complements of the following graphs: graphs with girth at least $5$, triangle-free Kneser graphs, and the Clebsch, Mesner, and Gewirtz graphs. This paper also highlights classes of graphs with $α(G) = 2$ where it is unknown if Hadwiger's Conjecture holds.

Figures (2)

• Figure 1: Relations between the conjectures. An arrow from one conjecture to another indicates that the former implies the latter.
• Figure 2: The graph $B_7$.

Theorems & Definitions (103)

• Conjecture 1
• Conjecture 2
• Theorem 1.1: DM82
• Theorem 1.2: BK11
• Conjecture 3: $\mathop{\mathrm{HC}}\nolimits_{\alpha = 2}$
• Conjecture 4: $\mathop{\mathrm{HC}}\nolimits_{n/2}$
• Conjecture 5: $\mathop{\mathrm{HC}}\nolimits_{\varepsilon}$
• Conjecture 6: Linear-CM
• Conjecture 7: $\text{4-CM}$
• Conjecture 8: $\mathop{\mathrm{SHC}}\nolimits_{\alpha = 2}$