The bunkbed conjecture is false (2410.02545v1)

Abstract: We give an explicit counterexample to the Bunkbed Conjecture introduced by Kasteleyn in 1985. The counterexample is given by a planar graph on $7222$ vertices, and is built on the recent work of Hollom (2024).

• The paper presents an explicit counterexample by constructing a complex planar graph with 7,222 vertices and 14,442 edges.
• It employs Hollom's framework to simulate hyperedge substitutions, revealing discrepancies in the expected connectivity probabilities.
• The findings challenge traditional percolation assumptions and call for a reexamination of network design and probabilistic analyses.

Overview of the Counterexample to the Bunkbed Conjecture

The paper presented by Nikita Gladkov, Igor Pak, and Aleksandr Zimin details an explicit counterexample to the Bunkbed Conjecture (BBC), a long-standing open problem in probability theory introduced by Kasteleyn in 1985. The authors construct a planar graph with 7222 vertices, relying on recent developments by Hollom, to demonstrate that the conjecture does not hold universally.

The Bunkbed Conjecture

The BBC posits the following: For a finite connected graph $G$ with a vertex set $V$, consider a "bunkbed" graph obtained by taking two copies of $G$ and connecting corresponding vertices within a specified subset $T$ using additional edges. The conjecture asserts that for any vertices $u, v \in V$, the probability that $u$ and $v$ are connected should be at least the probability that $u$ and the corresponding copy of $v$ are connected. This property, intuitively appealing and established in several specific cases, was speculative for general graphs.

Construction of the Counterexample

The authors disprove the conjecture by constructing a specific graph where the intended inequality does not hold. Their primary counterexample involves a complex planar graph that builds on hypergraph principles. By using Hollom's framework, which involves graph substitutions simulating hyperedges under varying probabilities, they establish a discrepancy between predicted connection probabilities.

Key Results and Implications

The paper's main result shows that there is a connected planar graph with specific connectivity properties, refuting the BBC. The graph contains:

• 7222 vertices
• 14442 edges

The authors establish that for particular configurations, the probability of connecting corresponding vertices violates the conjecture. This finding is consequential as it not only resolves a long-standing open problem but also suggests that reliance on the conjecture in probabilistic analyses must be revisited.

Theoretical and Practical Implications

The theoretical implications of this work are significant, as it revises our understanding of graph percolation properties. Practically, the construction demonstrates that care must be taken when using the BBC as a heuristic in network design, particularly in communications and statistical physics where large network structures are prevalent.

Future Directions

The authors hint at several avenues for future investigation:

• Exploring other special graph classes: Despite the refutation of the conjecture for general graphs, it may still hold for specific structured graphs or under additional conditions.
• Extensions to hypergraphs: Given the role of hypergraph theory in their construction, further explorations in hypergraph percolation and related models could yield new insights.
• Experimental validation and computational complexity: The challenge remains in computationally verifying such counterexamples due to the immense scale and probability differences involved, suggesting a need for more refined algorithms.

The paper sets a solid foundation for subsequent research into graph connectivity and opens up new questions in the exploration of probabilistic graph properties and their applications in theoretical and applied settings.