The maximum diameter of $d$-dimensional simplicial complexes

Abstract

For every fixed dimension $d$ and sufficiently large $n$, we determine the maximum possible diameter of a strongly connected $d$-dimensional simplicial complex on $n$ vertices. This improves on a sequence of previous results and settles a problem of Santos from 2013. On the way, as a special case, we also characterise the existence of an extra-tight Euler tour in the complete $d$-uniform hypergraph on $n$ vertices.

Figures (7)

• Figure 1: On the left: A strongly connected simplicial $2$-complex on $[9]$ with maximum diameter. The facets are given by the triangles. To ensure that the dual graph is a path, each pair in $\binom{[9]}{2}$ appears at most once as an edge. The diameter is maximum by \ref{['obs: size of F and its shadow']} because only the pair $\{4,7\}$ does not appear as an edge. On the right: A (non-optimal) straight simplicial $2$-complex on $[10]$.
• Figure 2: Overall proof structure
• Figure 3: The simplicial $d$-complex $\mathcal{C}'$ for $d=2$ and $d=3$.
• Figure 4: The inductive step using the Cover Down Lemma. By the inductive hypothesis, there is a (blue) extra-tight trail covering $G'\backslash G'[U_i]$. One end is extended by a (green) extra-tight trail into $U_{i+1}\backslash U_{i+2}$. Then, the Cover Down \ref{['lem: Cover Down']} gives us an (orange) extra-tight trail covering all remaining edges in $G'[U_i]\backslash G'[U_{i+1}]$. Finally, these two trails are connected via a short (red) extra-tight trail.
• Figure 5: Absorbing a cycle $C$ into the trail $A$. Vertices that appear in both $A$ and $C$ are shown in thicker ellipses. The new sequence is $(\ldots , w_0, u_1, u_2, \ldots , u_{d-1}, u_d, c_1 , \ldots , c_k, u_0, u_1, u_2, \ldots , u_{d-1}, w_d, \ldots )$.

Theorems & Definitions (94)

• Theorem 1.1
• proof
• Definition 1
• Definition 2
• Remark 1
• Definition 3
• Theorem 1.2
• Theorem 1.3
• Definition 4
• Definition 5