arXiv subjects

António Girão

Publications and source records attributed to António Girão.

At least 55 records · Page 3Linked to original sources

arXiv2021

Tiling with monochromatic bipartite graphs of bounded maximum degree

We prove that for any $r\in \mathbb{N}$, there exists a constant $C_r$ such that the following is true. Let $\mathcal{F}=\{F_1,F_2,\dots\}$ be an infinite sequence of bipartite graphs such that $|V(F_i)|=i$ and $Δ(F_i)\leq Δ$ hold for all $i$. Then in any $r$-edge coloured complete graph $K_n$, there is a collection of at most $\exp(C_rΔ)$ monochromatic subgraphs, each of which is isomorphic to an element of $\mathcal{F}$, whose vertex sets partition $V(K_n)$. This proves a conjecture of Corsten and Mendonça in a strong form and generalizes results on the multicolour Ramsey numbers of bounded-degree bipartite graphs.

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arXiv2021

Powers of paths and cycles in tournaments

We show that for every positive integer $k$, any tournament can be partitioned into at most $2^{ck}$ $k$-th powers of paths. This result is tight up to the exponential constant. Moreover, we prove that for every $\varepsilon>0$ and every integer $k$, any tournament on $n\ge \varepsilon^{-Ck}$ vertices which is $\varepsilon$-far from being transitive contains the $k$-th power of a cycle of length $Ω(\varepsilon n)$; both bounds are tight up to the implied constants.

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arXiv2021

Path and cycle decompositions of dense graphs

We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph on $n$ vertices can be decomposed into at most $\left\lceil \frac{n}{2}\right\rceil$ paths, while a conjecture of Hajós states that any Eulerian graph on $n$ vertices can be decomposed into at most $\left\lfloor \frac{n-1}{2}\right\rfloor$ cycles. The Erdős-Gallai conjecture states that any graph on $n$ vertices can be decomposed into $O(n)$ cycles and edges. We show that if $G$ is a sufficiently large graph on $n$ vertices with linear minimum degree, then the following hold. (i) $G$ can be decomposed into at most $\frac{n}{2}+o(n)$ paths. (ii) If $G$ is Eulerian, then it can be decomposed into at most $\frac{n}{2}+o(n)$ cycles. (iii) $G$ can be decomposed into at most $\frac{3 n}{2}+o(n)$ cycles and edges. If in addition $G$ satisfies a weak expansion property, we asymptotically determine the required number of paths/cycles for each such $G$. (iv) $G$ can be decomposed into $\max \left\{\frac{odd(G)}{2},\frac{Δ(G)}{2}\right\}+o(n)$ paths, where $odd(G)$ is the number of odd-degree vertices of $G$. (v) If $G$ is Eulerian, then it can be decomposed into $\frac{Δ(G)}{2}+o(n)$ cycles. All bounds in (i)-(v) are asymptotically best possible.

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arXiv2021

Powers of paths in tournaments

In this short note we prove that every tournament contains the $k$-th power of a directed path of linear length. This improves upon recent results of Yuster and of Girão. We also give a complete solution for this problem when $k=2$, showing that there is always a square of a directed path of length $\lceil 2n/3 \rceil-1$, which is best possible.

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arXiv2020

Strong complete minors in digraphs

Kostochka and Thomason independently showed that any graph with average degree $Ω(r\sqrt{\log r})$ contains a $K_r$ minor. In particular, any graph with chromatic number $Ω(r\sqrt{\log r})$ contains a $K_r$ minor, a partial result towards Hadwiger's famous conjecture. In this paper, we investigate analogues of these results in the directed setting. There are several ways to define a minor in a digraph. One natural way is as follows. A strong $\overrightarrow{K}_r$ minor is a digraph whose vertex set is partitioned into $r$ parts such that each part induces a strongly-connected subdigraph, and there is at least one edge in each direction between any two distinct parts. We investigate bounds on the dichromatic number and minimum out-degree of a digraph that force the existence of strong $\overrightarrow{K}_r$ minors as subdigraphs. In particular, we show that any tournament with dichromatic number at least $2r$ contains a strong $\overrightarrow{K}_r$ minor, and any tournament with minimum out-degree $Ω(r\sqrt{\log r})$ also contains a strong $\overrightarrow{K}_r$ minor. The latter result is tight up to the implied constant, and may be viewed as a strong-minor analogue to the classical result of Kostochka and Thomason. Lastly, we show that there is no function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that any digraph with minimum out-degree at least $f(r)$ contains a strong $\overrightarrow{K}_r$ minor, but such a function exists when considering dichromatic number.

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arXiv2020

A note on long powers of paths in tournaments

A square of a path on $k$ vertices is a directed path $x_1\ldots x_k$, where $x_i$ is directed to $x_{i+2}$, for every $i\in \{1,\ldots, k-1\}$. Recently, Yuster showed that any tournament on $n$ vertices contains a square of a path of length at least $n^{0.295}$. In this short note, we improve this bound. More precisely, we show that for every $\varepsilon>0$, there exists $c_{\varepsilon}>0$ such that any tournament on $n$ vertices contains a square of a path on at least $c_{\varepsilon}n^{1-\varepsilon}$ vertices.

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arXiv2020

Dirac's theorem for random regular graphs

We prove a `resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\varepsilon>0$, a.a.s. the following holds: let $G'$ be any subgraph of the random $n$-vertex $d$-regular graph $G_{n,d}$ with minimum degree at least $(1/2+\varepsilon)d$. Then $G'$ is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that $d$ is large cannot be omitted, and secondly, the minimum degree bound cannot be improved.

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arXiv2020

A Canonical Polynomial Van der Waerden's Theorem

We prove a canonical polynomial Van der Waerden's Theorem. More precisely, we show the following. Let $\{p_1(x),\ldots,p_k(x)\}$ be a set of polynomials such that $p_i(x)\in \mathbb{Z}[x]$ and $p_i(0)=0$, for every $i\in \{1,\ldots,k\}$. Then, in any colouring of $\mathbb{Z}$, there exist $a,d\in \mathbb{Z}$ such that $\{a+p_1(d),\ldots,a+p_{k}(d)\}$ forms either a monochromatic or a rainbow set.

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arXiv2020

Subgraphs of large connectivity and chromatic number

Resolving a problem raised by Norin, we show that for each $k \in \mathbb{N}$, there exists an $f(k) \le 7k$ such that every graph $G$ with chromatic number at least $f(k)+1$ contains a subgraph $H$ with both connectivity and chromatic number at least $k$. This result is best-possible up to multiplicative constants, and sharpens earlier results of Alon-Kleitman-Thomassen-Saks-Seymour from 1987 showing that $f(k) = O(k^3)$, and of Chudnovsky-Penev-Scott-Trotignon from 2013 showing that $f(k) = O(k^2)$. Our methods are robust enough to handle list colouring as well: we also show that for each $k \in \mathbb{N}$, there exists an $f_\ell(k) \le 4k$ such that every graph $G$ with list chromatic number at least $f_\ell(k)+1$ contains a subgraph $H$ with both connectivity and list chromatic number at least $k$. This result is again best-possible up to multiplicative constants; here, unlike with $f(\cdot)$, even the existence of $f_\ell(\cdot)$ appears to have been previously unknown.

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arXiv2020

On Covering Numbers, Young Diagrams, and the Local Dimension of Posets

We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular we show that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles, and prove that this is best-possible. This answers two questions by Kim, Martin, Masa{ř}{\'ı}k, Shull, Smith, Uzzell, and Wang (Europ. J. Comb. 2020), namely: - What is the local complete bipartite cover number of a difference graph? - Is there a sequence of graphs with constant local difference graph cover number and unbounded local complete bipartite cover number? We add to the study of these local covering numbers with a lower bound construction and some examples. Following Kim \emph{et al.}, we use the results on local covering numbers to provide lower and upper bounds for the local dimension of partially ordered sets of height~2. We discuss the local dimension of some posets related to Boolean lattices and show that the poset induced by the first two layers of the Boolean lattice has local dimension $(1 + o(1))\log_2\log_2 n$. We conclude with some remarks on covering numbers for digraphs and Ferrers dimension.

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arXiv2019

$(2k+1)$-connected tournaments with large minimum out-degree are $k$-linked

Pokrovskiy conjectured that there is a function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that any $2k$-strongly-connected tournament with minimum out and in-degree at least $f(k)$ is $k$-linked. In this paper, we show that any $(2k+1)$-strongly-connected tournament with minimum out-degree at least some polynomial in $k$ is $k$-linked, thus resolving the conjecture up to the additive factor of $1$ in the connectivity bound, but without the extra assumption that the minimum in-degree is large. Moreover, we show the condition on high minimum out-degree is necessary by constructing arbitrarily large tournaments that are $(2.5k-1)$-strongly-connected but are not $k$-linked.

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arXiv2019

Rainbow saturation of graphs

In this paper we study the following problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger. Given a graph $H$ and an integer $t$, what is $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(H)}\right)$, the minimum number of edges in a $t$-edge-coloured graph $G$ on $n$ vertices such that $G$ does not contain a rainbow copy of $H$, but adding to $G$ a new edge in any colour from $\{1,2,\ldots,t\}$ creates a rainbow copy of $H$? Here, we completely characterize the growth rates of $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(H)}\right)$ as a function of $n$, for any graph $H$ belonging to a large class of connected graphs and for any $t\geq e(H)$. This classification includes all connected graphs of minimum degree $2$. In particular, we prove that $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(K_r)}\right)=Θ(n\log n)$, for any $r\geq 3$ and $t\geq {r \choose 2}$, thus resolving a conjecture of Barrus, Ferrara, Vandenbussche, and Wenger. We also pose several new problems and conjectures.

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arXiv2019

A large number of $m$-coloured complete infinite subgraphs

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is $m$-\textit{coloured} if each of the $m$ colours is used. For an $m$-colouring $Δ$ of $\mathbb{N}^{(2)}$, the complete graph on $\mathbb{N}$, we denote by $\mathcal{F}_Δ$ the set all values $γ$ for which there exists an infinite subset $X\subset \mathbb{N}$ such that $X^{(2)}$ is $γ$-coloured. Properties of this set were first studied by Erickson in $1994$. Here, we are interested in estimating the minimum size of $\mathcal{F}_Δ$ over all $m$-colourings $Δ$ of $\mathbb{N}^{(2)}$. Indeed, we shall prove the following result. There exists an absolute constant $α> 0$ such that for any positive integer $m \neq \left\{ {n \choose 2}+1, {n \choose 2}+2: n\geq 2\right\}$, $|\mathcal{F}_Δ| \geq (1+α)\sqrt{2m}$, for any $m$-colouring $Δ$ of $\mathbb{N}^{(2)}$, thus proving a conjecture of Narayanan. This result is tight up to the order of the constant $α$.

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arXiv2019

Disjoint pairs in set systems with restricted intersection

The problem of bounding the size of a set system under various intersection restrictions has a central place in extremal combinatorics. We investigate the maximum number of disjoint pairs a set system can have in this setting. In particular, we show that for any pair of set systems $(\mathcal{A}, \mathcal{B})$ which avoid a cross-intersection of size $t$, the number of disjoint pairs $(A, B)$ with $A \in \mathcal{A}$ and $B \in \mathcal{B}$ is at most $\sum_{k=0}^{t-1}\binom{n}{k}2^{n-k}$. This implies an asymptotically best possible upper bound on the number of disjoint pairs in a single $t$-avoiding family $\mathcal{F} \subset \mathcal{P}[n]$. We also study this problem when $\mathcal{A}$, $\mathcal{B} \subset [n]^{(r)}$ are both $r$-uniform, and show that it is closely related to the problem of determining the maximum of the product $|\mathcal{A}||\mathcal{B}|$ when $\mathcal{A}$ and $\mathcal{B}$ avoid a cross-intersection of size $t$, and $n \ge n_0(r, t)$.

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arXiv2019

Highly linked tournaments with large minimum out-degree

We prove that there exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integer $k$, if $T$ is a strongly $4k$-connected tournament with minimum out-degree at least $f(k)$, then $T$ is $k$-linked. This makes progress towards resolving a conjecture of Pokrovskiy. Along the way, we show that a tournament with sufficiently large minimum out-degree contains a subdivision of a complete directed graph. This result may be of independent interest.

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arXiv2019

Subdivisions of digraphs in tournaments

We show that for every positive integer $k$, any tournament with minimum out-degree at least $(2+o(1))k^2$ contains a subdivision of the complete directed graph on $k$ vertices, which is best possible up to a factor of $8$. This may be viewed as a directed analogue of a theorem proved by Bollobás and Thomason, and independently by Komlós and Szemerédi, concerning subdivisions of cliques in graphs with sufficiently high average degree. We also consider the following problem: given $k$, what is the smallest positive integer $f(k)$ such that any $f(k)$-vertex tournament contains a $1$-subdivision of the transitive tournament on $k$ vertices? We show that $f(k)= O\left (k^2\log^3 k\right)$ which is best possible up to the logarithmic factors.

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arXiv2019

Turán theorems for unavoidable patterns

We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and by Fox and Sudakov. First, for $t \ge 3$, we show that any two-colouring of the complete graph on $n$ vertices that is $δ$-far from being monochromatic contains an \emph{unavoidable $t$-colouring} when $δ\gg n^{-1/t}$, where an unavoidable $t$-colouring is any two-colouring of a clique of order $2t$ in which one colour forms either a clique of order $t$ or two disjoint cliques of order $t$. Next, for $ t\ge 3$, we show that any tournament on $n$ vertices that is $δ$-far from being transitive contains an \emph{unavoidable $t$-tournament} when $δ\gg n^{-1/\lceil t/2 \rceil}$, where an unavoidable $t$-tournament is the blow-up of a cyclic triangle obtained by replacing each vertex of the triangle by a transitive tournament of order $t$. Conditional on a well-known conjecture about bipartite Turán numbers, both results are sharp up to implied constants and hence determine the order of magnitude of the corresponding off-diagonal Ramsey numbers.

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arXiv2018

A precolouring extension of Vizing's theorem

Fix a palette $\mathcal K$ of $Δ+1$ colours, a graph with maximum degree $Δ$, and a subset $M$ of the edge set with minimum distance between edges at least $9$. If the edges of $M$ are arbitrarily precoloured from $\mathcal K$, then there is guaranteed to be a proper edge-colouring using only colours from $\mathcal K$ that extends the precolouring on $M$ to the entire graph. This result is a first general precolouring extension form of Vizing's theorem, and it proves a conjecture of Albertson and Moore under a slightly stronger distance requirement. We also show that the condition on the distance can be lowered to $5$ when the graph contains no cycle of length $5$.

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