arXiv subjects

Silas Rathke

Publications and source records attributed to Silas Rathke.

5 recordsLinked to original sources

arXiv2026

No extremal square-free words over alphabets of size at least 5

A word over an alphabet $\mathbb A$ contains a square if it has a subword of the form $XX$ where $X$ is a word. A word $W$ is \emph{extremal square-free} if it does not contain a square, but it contains a square as soon as any letter of $\mathbb A$ is inserted at any position of $W$. Grytczuk, Kordulewski, and Niewiadomski conjectured that there are no extremal square-free words over alphabets of size at least 4. We prove this for alphabets of size at least 5. Our proof also implies that the sequence of \emph{nonchalant words} defined by Grytczuk, Kordulewski, and Niewiadomski is infinite and converges to an infinite word for all alphabets of size at least 5.

math.CO↗

arXiv2026

The maximum diameter of $d$-dimensional simplicial complexes

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.

math.CO↗

arXiv2025

Spanning Components and Surfaces Under Minimum Vertex Degree

We study minimum vertex-degree conditions in 3-uniform hypergraphs for (tight) spanning components and (combinatorial) surfaces. Our main results show that a 3-uniform hypergraph $G$ on $n$ vertices contains a spanning component if $δ_1(G) \gtrsim \tfrac{1}{2} \binom{n}{2}$ and a spanning copy of any surface if $δ_1(G) \gtrsim \tfrac{5}{9} \binom{n}{2}$, which in both cases is asymptotically optimal. This extends the work of Georgakopoulos, Haslegrave, Montgomery, and Narayanan who determined the corresponding minimum codegree conditions in this setting.

math.CO↗

arXiv2025

The maximum diameter of 2-dimensional simplicial complexes

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.

math.CO↗

arXiv2024

On the chromatic number of powers of subdivisions of graphs

For a given graph $G=(V,E)$, we define its \emph{$n$th subdivision} as the graph obtained from $G$ by replacing every edge by a path of length $n$. We also define the \emph{$m$th power} of $G$ as the graph on vertex set $V$ where we connect every pair of vertices at distance at most $m$ in $G$. In this paper, we study the chromatic number of powers of subdivisions of graphs and resolve the case $m=n$ asymptotically. In particular, our result confirms a conjecture of Mozafari-Nia and Iradmusa in the case $m=n=3$ in a strong sense.

math.CO↗