What the study found
The study found the exact maximum possible diameter for 2-dimensional simplicial complexes on n vertices. A simplicial complex is a mathematical object built from vertices, edges, and higher-dimensional faces.
Why the authors say this matters
The authors say this addresses a problem posed by Santos about the largest possible diameter of a d-dimensional simplicial complex on n vertices. The findings also point to an open problem about packing squares of Hamilton cycles in the complete graph.
What the researchers tested
The researchers studied the maximum diameter problem for abstract simplicial complexes, focusing on dimension 2. They used an explicit construction and also obtained a sequence of explicit constructions that are tight.
What worked and what didn't
For dimension 2, they determined the exact maximum for every n. They also came across an open problem about packing squares of Hamilton cycles in the complete graph, which is not resolved in the abstract.
What to keep in mind
The abstract only states results for dimension 2, not for all dimensions. It does not describe limitations beyond the mention of an open problem about Hamilton cycle squares.
Key points
- The exact maximum diameter was determined for 2-dimensional simplicial complexes on n vertices.
- The result was obtained using an explicit construction.
- The paper addresses a problem posed by Santos about the largest possible diameter of a simplicial complex.
- The authors also found an open problem involving packing squares of Hamilton cycles in the complete graph.
- They report an infinite sequence of tight explicit constructions.
Disclosure
- Research title:
- Exact maximum diameter determined for 2-dimensional simplicial complexes
- Authors:
- Olaf Parczyk, Silas Rathke, Tibor Szabó
- Institutions:
- Freie Universität Berlin, Freie Universität Berlin, Freie Universität Berlin
- Publication date:
- 2026-04-27
- OpenAlex record:
- View
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