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The maximum diameter of 2-dimensional simplicial complexes

Olaf Parczyk, Silas Rathke, Tibor Szabó

TL;DR

This work determines the exact maximal diameter $H_s(n,2)$ of connected dual graphs of 2-dimensional simplicial complexes on $n$ vertices for all $n\neq 6$ by constructing explicit circular-good-sequences of triangles (using generating sequences) that achieve the upper bound $H_s(n,2)\le \left\lfloor\frac{1}{2}\binom{n}{2}-\frac{3}{2}\right\rfloor$. It develops a comprehensive encoding of simplicial 2-complexes, introduces generating sequences, and provides residue-based constructions to realize the maximum diameter for each $n$ in the four residue classes mod $4$, including $n=4k+1$, $n=4k+4$, $n=4k+3$, and $n=4k+6$, with small-$n$ cases treated in an appendix. Beyond the $n$-vertex result, the paper proves an explicit decomposition of $E(K_p)$ into squares of Hamilton cycles for infinitely many primes $p$ (and near-complete packings for a wide range of $n$), obtaining improved asymptotics for packing problems. The concluding discussion broadens the impact to $d\ge 3$ via a partition/cut/glue strategy yielding a near-optimal lower bound on $H_s(n,d)$, connects to $2$-radius sequences and harmonious colorings, and situates the results within the history of the Hirsch-type diameter problems and related constructions.

Abstract

We study a problem of Santos about the largest possible diameter of a $d$-dimensional (abstract) simplicial complex on $n$ vertices. For dimension 2, we determine the exact value of the maximum for every $n$ using an explicit construction. We also come across a tantalizing open problem about the packing of squares of Hamilton cycles in the complete graph and obtain an infinite sequence of tight explicit constructions.

The maximum diameter of 2-dimensional simplicial complexes

TL;DR

This work determines the exact maximal diameter of connected dual graphs of 2-dimensional simplicial complexes on vertices for all by constructing explicit circular-good-sequences of triangles (using generating sequences) that achieve the upper bound . It develops a comprehensive encoding of simplicial 2-complexes, introduces generating sequences, and provides residue-based constructions to realize the maximum diameter for each in the four residue classes mod , including , , , and , with small- cases treated in an appendix. Beyond the -vertex result, the paper proves an explicit decomposition of into squares of Hamilton cycles for infinitely many primes (and near-complete packings for a wide range of ), obtaining improved asymptotics for packing problems. The concluding discussion broadens the impact to via a partition/cut/glue strategy yielding a near-optimal lower bound on , connects to -radius sequences and harmonious colorings, and situates the results within the history of the Hirsch-type diameter problems and related constructions.

Abstract

We study a problem of Santos about the largest possible diameter of a -dimensional (abstract) simplicial complex on vertices. For dimension 2, we determine the exact value of the maximum for every using an explicit construction. We also come across a tantalizing open problem about the packing of squares of Hamilton cycles in the complete graph and obtain an infinite sequence of tight explicit constructions.

Paper Structure

This paper contains 22 sections, 12 theorems, 61 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main, 4k+1']}
  3. Encoding simplicial 2-complexes
  4. Generating sequences and circular simplicial complexes
  5. The Construction for $n=4k+1$
  6. The Construction for $n=4k+4$
  7. The Construction for $n=4k+3$
  8. The Construction for $n=4k+6$
  9. Proof of Theorem \ref{['thm:packHC2']}
  10. Concluding remarks
  11. Packing squares of Hamilton cycles.
  12. Dimension $d\geq 3$.
  13. History of the problem and pseudo-manifolds.
  14. Relation to $2$-radius sequences and harmonious colourings.
  15. Note.
  16. ...and 7 more sections

Key Result

Theorem 1.1

For $n\ge 3$, it holds that

Figures (12)

  • Figure 1: The black sequence of triangles represents a simplicial 2-complex of diameter 7 for $n=7$. We cannot add the red triangle to it to create a simplicial 2-complex of diameter 8 because all edges containing $1$ or $4$ are already used.
  • Figure 2: Simplicial complex for $n=9$.
  • Figure 3: Circular simplicial complex for $n=13$. The first and the last triangles are glued together via the edge $\lbrace0,1\rbrace$. It can be cut at any blue edge to create an optimal simplicial complex.
  • Figure 4: Circular simplicial complex for $n=17$ corresponding to the generating sequence $3\overset{\textcolor{blue}{\space5\space}}{~}2\overset{\textcolor{blue}{\space9\space}}{~}7\overset{\textcolor{blue}{\space13\space}}{~}\widehat{6}\overset{\textcolor{blue}{\space16\space}}{~}$
  • Figure 5: Simplicial complex for $n=16$. Only nine edges $\lbrace 0,7\rbrace$, $\lbrace 1,7\rbrace$, $\lbrace 1,8\rbrace$, $\lbrace 1,10\rbrace$, $\lbrace 3,8\rbrace$, $\lbrace 7,a\rbrace$, $\lbrace8,c\rbrace$, $\lbrace a,c\rbrace$ and $\lbrace b,c\rbrace$ are missing.
  • ...and 7 more figures

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.3
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof : Proof of Proposition \ref{['prop: with turn']}.
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 19 more