Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.03603

The vertex sets of subtrees of a tree

Maria Chudnovsky·Tung Nguyen·Alex Scott·Paul Seymour
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $\mathcal{F}$ be a set of subsets of a set $W$. When is there a tree $T$ with vertex set $W$ such that each member of $\mathcal{F}$ is the set of vertices of a subtree of $T$? It is necessary that $\mathcal{F}$ has the Helly property and the intersection graph of $\mathcal{F}$ is chordal. We will show that these two necessary conditions are together sufficient in the finite case, and more generally, they are sufficient if no element of $W$ belongs to infinitely many infinite sets in $\mathcal{F}$.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Maria Chudnovsky, Tung Nguyen, Alex Scott, Paul Seymour. 2025-06-04. The vertex sets of subtrees of a tree. https://arxiv.org/abs/2506.03603

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On nut graphs with two vertex and three edge orbits

Nut graphs are graphs whose adjacency matrix is singular with one-dimensional null space spanned by a vector with no zero entries. In a recent paper, Bašić, Fowler and Pisanski proved that the automorphism group of a nut graph has more orbits on the edge set than on the vertex set. They classified all orders for which a vertex-transitive nut graph with precisely two edge orbits exists, and conjectured that a nut graph with two vertex and three edge orbits exists for each non-prime order $n \ge 9$. Motivated by this conjecture, we introduce a very general construction that provides graphs with the desired symmetry properties, and we determine some sufficient spectral and structural conditions under which they are nut graphs. The construction yields infinite families of examples and confirms the above conjecture for all odd non-prime orders up to $2\,500$ and for at least $99.8$ percent of all odd non-prime orders up to a million. Finally, we present some additional interesting examples of nut graphs with two vertex and three edge orbits that do not arise from this construction.

math.CO↗
arXiv2026

On vertex-minimal simplicial maps to the sphere

For positive integers $n,d$, let $λ(n,d)$ be the minimal number of vertices of a triangulation of the $n$-sphere which admits a degree $d$ simplicial map onto the boundary of the $(n+1)$-simplex. We show that for $h=\lfloor\frac{n+1}2\rfloor$, the function $λ(n,d)^h$ has linear order of growth in $d$, answering a question of O. Musin. All triangulations we obtained are isomorphic to boundaries of convex polytopes in $\mathbb{R}^{n+1}$.

math.CO↗