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Giovanne Santos

Publications and source records attributed to Giovanne Santos.

8 recordsLinked to original sources
arXiv2026

Are trees really just butterflies in disguise?

As a generalisation of the Erdős-Sós conjecture about graphs, Addario-Berry, Havet, Linhares Sales, Reed and Thomassé conjectured that every digraph on $n$ vertices with more than $(k-1)n$ arcs contains every antidirected tree with $k$ arcs. We prove a dense, approximate version of this for trees with bounded maximum degree, as well as for trees whose layers are evenly distributed. We use a regularity based approach, centred around finding a copy of a given tree in the blow up of a caterpillar.

math.CO↗
arXiv2026

Semidegree threshold for spanning trees in oriented graphs

We show that for all $γ> 0$ and $Δ\in \mathbb{N}$, there is some $n_0$ such that, if $n \geq n_0$, then every oriented graph on $n$ vertices with minimum semidegree at least $(3/8 + γ)n$ contains a copy of each oriented tree on $n$ vertices with maximum degree at most $Δ$. This is asymptotically best possible.

math.CO↗
arXiv2026

Canonical Ramsey theorem for graphs with clean intersections

Extending earlier results of Nešetřil and Rödl [Selective graphs and hypergraphs, Ann. Discrete Math. 3 (1978), 181--189], we show that for every ordered graph $F$ there exist an ordered graph $H$ and a system $\mathscr{H}_F$ of induced copies of $F$ such that every colouring of the edges of $H$ yields a canonically coloured copy of $F$ from $\mathscr{H}_F$ and any two copies from $\mathscr{H}_F$ intersect either in a vertex or an edge or not at all. As a consequence, this allows us to construct, for any given ordered graph $F$, canonical Ramsey graphs $H$ enjoying additional structural properties. In particular, $H$ can have the same clique number as $F$ and, provided $F$ is not bipartite, the same odd girth. Moreover, if $F$ is connected, then the copies of $F$ from $\mathscr{H}_F$ are not only induced, but their pairs of vertices also have the same distances in $H$ as in $F$.

math.CO↗
arXiv2025

The Brown-Erdős-Sós conjecture in dense triple systems

The famous Brown-Erdős-Sós conjecture from 1973 states, in an equivalent form, that for any fixed $δ>0$ and integer $k\geq 3$ every sufficiently large linear $3$-uniform hypergraph of size $δn^2$ contains some $k$ edges spanning at most $k+3$ vertices. We prove it to hold for $δ>4/5$, establishing the first bound of this kind.

math.CO↗
arXiv2025

Antidirected trees in directed graphs

The Komlós-Sárközy-Szemerédi (KSS) theorem establishes that a certain bound on the minimum degree of a graph guarantees it contains all bounded degree trees of the same order. Recently several authors put forward variants of this result, where the tree is of smaller order than the host graph, and the host graph also obeys a maximum degree condition. Also, Kathapurkar and Montgomery extended the KSS theorem to digraphs. We bring these two directions together by establishing minimum and maximum degree bounds for digraphs that ensure the containment of oriented trees of smaller order. Our result is restricted to balanced antidirected trees of bounded degree. More precisely, we show that for every $γ> 0$, $c\in\mathbb{R}$, $\ell\geq 2$ sufficiently large $n$ and all $k\geqγn$, the following holds for every $n$-vertex digraph $D$ and every balanced antidirected tree $T$ with $k$ arcs whose total maximum degree is bounded by $(\log n)^c$. If $D$ has a vertex of outdegree at least $(1+γ)(\ell -1)k$, a vertex of indegree at least $(1+γ)(\ell -1)k$ and minimum semidegree $δ^0(D)\geq\left(\frac{\ell}{2\ell -1}+γ\right)k$, then $D$ contains $T$.

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arXiv2024

A study on token digraphs

For a digraph $D$ of order $n$ and an integer $1 \leq k \leq n-1$, the $k$-token digraph of $D$ is the graph whose vertices are all $k$-subsets of vertices of $D$ and, given two such $k$-subsets $A$ and $B$, $(A,B)$ is an arc in the $k$-token digraph whenever $\{a\} = A \setminus B$, $\{b\} = B \setminus A$, and there is an arc $(a,b)$ in $D$. Token digraphs are a generalization of token graphs. In this paper, we study some properties of token digraphs, including strong and unilateral connectivity, kernels, girth, circumference and Eulerianity. We also extend some known results on the clique and chromatic numbers of $k$-token graphs, addressing the bidirected clique number and dichromatic number of $k$-token digraphs. Additionally, we prove that determining whether $2$-token digraphs have a kernel is NP-complete.

math.CO↗
arXiv2024

Packing large balanced trees into bipartite graphs

We prove that for every ${γ> 0}$ there exists $n_0 \in \mathbb{N}$ such that for every ${n \geq n_0}$ any family of up to $\lfloor{n^{\frac12+γ}}\rfloor$ trees having at most $(1-γ)n$ vertices in each bipartition class can be packed into $K_{n,n}$. As a tool for our proof, we show an approximate bipartite version of the Komlós-Sárközy-Szemerédi Theorem, which we believe to be of independent interest.

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