arXiv subjects

António Girão

Publications and source records attributed to António Girão.

At least 19 recordsLinked to original sources

arXiv2026

Cycle-factors of regular graphs via entropy

It is a classical result that a random permutation of $n$ elements has, on average, about $\log n$ cycles. We generalise this fact to all directed $d$-regular graphs on $n$ vertices by showing that, on average, a random cycle-factor of such a graph has $\mathcal{O}((n\log d)/d)$ cycles. This is tight up to the constant factor and improves the best previous bound of the form $\mathcal{O}(n/\sqrt{\log d})$ due to Vishnoi. Our results also yield randomised polynomial-time algorithms for finding such a cycle-factor and for finding a tour of length $(1+\mathcal{O}((\log d)/d)) \cdot n$ if the graph is connected. This makes progress on a conjecture of Magnant and Martin and on a problem studied by Vishnoi and by Feige, Ravi, and Singh. Our proof uses the language of entropy to exploit the fact that the upper and lower bounds on the number of perfect matchings in regular bipartite graphs are extremely close.

math.CO↗

arXiv2026

Two Relaxations of the Dominating Hadwiger's Conjecture

Illingworth and Wood recently proposed the Dominating Hadwiger's Conjecture, a strengthening of Hadwiger's Conjecture which asserts that every graph with no dominating $K_t$-model is $(t-1)$-colorable. We prove two relaxations of this conjecture. First, we show that every graph with average degree $Ct (\log t)^2$ contains a dominating $K_t$-model for some absolute constant $C$. This bound improves on the $2^{t-2}$ due to Illingworth and Wood and is within an $O(\log t)$ factor from optimal. Second, we prove that the vertices of every graph with no dominating $K_t$-model can be partitioned into $t-1$ parts such that the subgraph induced by each part has bounded maximum degree.

math.CO↗

arXiv2026

Non-Homotopic Drawings of Multigraphs

A multigraph drawn in the plane is non-homotopic if no two edges connecting the same pair of vertices can be continuously deformed into each other without passing through a vertex, and is $k$-crossing if every pair of edges (self-)intersects at most $k$ times. We prove that the number of edges in an $n$-vertex non-homotopic $k$-crossing multigraph is at most $6^{13 n (k + 1)}$, which is a substantial improvement over previous upper bounds. We also study this problem in the setting of monotone drawings where every edge is an x-monotone curve. We show that the number of edges, $m$, in such a drawing is at most $2 \binom{2n}{k + 1}$ and the number of crossings is $Ω\bigl(\frac{m^{2 + 1/k}}{n^{1 + 1/k}}\bigr)$. For fixed $k$ these bounds are both best possible up to a constant multiplicative factor.

math.CO↗

arXiv2026

The Dominating 4-Colour Theorem

A "dominating $K_t$-model" in a graph $G$ is a sequence $(T_1,\dots,T_t)$ of pairwise vertex-disjoint connected subgraphs of $G$, such that whenever $1\leq i<j\leq t$ every vertex in $T_j$ has a neighbour in $T_i$. Replacing "every vertex in $T_j$" by "some vertex in $T_j$" retrieves the standard definition of $K_t$-model, which is equivalent to a $K_t$-minor in $G$. We prove that every graph with no dominating $K_5$-model is $4$-colourable. This generalises and is significantly stronger than the 4-colour theorem for planar graphs or for graphs with no $K_5$-minor. It also makes progress towards Hajós' conjecture on $K_5$-subdivisions in $5$-chromatic graphs.

math.CO↗

arXiv2026

On $2$-factors of Hamiltonian graphs

Let $k\geq 2$. We show that, for a sufficiently small $\varepsilon>0$, any sufficiently large $n$-vertex Hamiltonian graph of minimum degree at least $n^{1-\varepsilon}$ contains a $2$-factor consisting of exactly $k$ cycles. This is the first minimum-degree condition which is polynomially smaller than linear. Our methods yield an analogous result when the host graph is not required to contain a Hamilton cycle, but only a $2$-factor consisting of at most $k$ cycles; this answers a question of Bucić, Jahn, Pokrovskiy and Sudakov.

math.CO↗

arXiv2026

Induced subdivisions of $K_{d+1}$ in graphs of high girth

In this paper, we show that for all $k\geq 10^8$, every graph with minimum degree $k$ and girth at least $10^8$ contains an induced subdivision of a $K_{k+1}$. This answers a problem asked by Kühn and Osthus (originally attributed to Shi).

math.CO↗

arXiv2026

Cycles with almost linearly many chords

We prove that constant minimum degree already forces cycles with almost linearly many chords. Specifically, every graph $G$ with $δ(G)\ge C$ contains a cycle of length $\ell\ge 4$ with $Ω(\ell/\log^{C}\ell)$ chords for some absolute constant $C>0$. This is the first result showing that a constant-degree condition yields an unbounded -- indeed nearly linear -- number of chords, placing our bound within a polylogarithmic factor of the Chen--Erdős--Staton conjecture. It also gives a strong affirmative conclusion in the direction of a recent question of Dvořák, Martins, Thomassé, and Trotignon asking whether constant-degree graphs must contain cycles whose chord counts grow with their length.

math.CO↗

arXiv2025

Blowups of triangle-free graphs

A highly influential result of Nikiforov states that if an $n$-vertex graph $G$ contains at least $γn^h$ copies of a fixed $h$-vertex graph $H$, then $G$ contains a blowup of $H$ of order $Ω_{γ,H}(\log n)$. While the dependence on $n$ is optimal, the correct dependence on $γ$ is unknown; all known proofs yield bounds that are polynomial in $γ$, but the best known upper bound, coming from random graphs, is only logarithmic in $γ$. It is a major open problem to narrow this gap. We prove that if $H$ is triangle-free, then the logarithmic behavior of the upper bound is the truth. That is, under the assumptions above, $G$ contains a blowup of $H$ of order $Ω_H (\log n/{\log(1/γ)})$. This is the first non-trivial instance where the optimal dependence in Nikiforov's theorem is known. As a consequence, we also prove an upper bound on multicolor Ramsey numbers of blowups of triangle-free graphs, proving that the dependence on the number of colors is polynomial once the blowup is sufficiently large. This shows that, from the perspective of multicolor Ramsey numbers, blowups of fixed triangle-free graphs behave like bipartite graphs.

math.CO↗

arXiv2025

Measurable one-ended spanning trees

We show that a one-ended, locally finite, measurable graph on a standard probability space admits a measurable one-ended spanning subtree if and only if it is measure-hyperfinite. This answers a question posed by Bowen, Poulin, and Zomback and extends recent results of Timár and Conley, Gaboriau, Marks, and Tucker-Drob.

math.LO↗

arXiv2025

New bounds for linear arboricity and related problems

A linear forest is a collection of vertex-disjoint paths. The Linear Arboricity Conjecture states that every graph of maximum degree $Δ$ can be decomposed into at most $\lceil(Δ+1)/2\rceil$ linear forests. We prove that $Δ/2 + \mathcal{O}(\log n)$ linear forests suffice, where $n$ is the number of vertices of the graph. If $Δ= Ω(n^\varepsilon)$, this is an exponential improvement over the previous best error term. We achieve this by generalising Pósa rotations from rotations of one endpoint of a path to simultaneous rotations of multiple endpoints of a linear forest. This method has further applications, including the resolution of a conjecture of Feige and Fuchs on spanning linear forests with few paths and the existence of optimally short tours in connected regular graphs.

math.CO↗

arXiv2025

Partitioning a tournament into sub-tournaments of high connectivity

We prove that there exists a constant $c > 0$ such that the vertices of every strongly $c \cdot kt$-connected tournament can be partitioned into $t$ parts, each of which induces a strongly $k$-connected tournament. This is clearly tight up to a constant factor, and it confirms a conjecture of Kühn, Osthus and Townsend (2016).

math.CO↗

arXiv2025

A bounded diameter strengthening of Kőnig's Theorem

K\H onig's theorem says that the vertex cover number of every bipartite graph is at most its matching number (in fact they are equal since, trivially, the matching number is at most the vertex cover number). An equivalent formulation of K\H onig's theorem is that in every $2$-colouring of the edges of a graph $G$, the number of monochromatic components needed to cover the vertex set of $G$ is at most the independence number of $G$. We prove the following strengthening of K\H onig's theorem: In every $2$-colouring of the edges of a graph $G$, the number of monochromatic subgraphs of bounded diameter needed to cover the vertex set of $G$ is at most the independence number of $G$.

math.CO↗

arXiv2024

A multidimensional Ramsey Theorem

Ramsey theory is a central and active branch of combinatorics. Although Ramsey numbers for graphs have been extensively investigated since Ramsey's work in the 1930s, there is still an exponential gap between the best known lower and upper bounds. For $k$-uniform hypergraphs, the bounds are of tower-type, where the height grows with $k$. Here, we give a multidimensional generalisation of Ramsey's Theorem to Cartesian products of graphs, proving that a doubly exponential upper bound suffices in every dimension. More precisely, we prove that for every positive integers $r,n,d$, in any $r$-colouring of the edges of the Cartesian product $\square^{d} K_N$ of $d$ copies of $K_N$, there is a copy of $\square^{d} K_n$ such that the edges in each direction are monochromatic, provided that $N\geq 2^{2^{C_drn^{d}}}$. As an application of our approach we also obtain improvements on the multidimensional Erdős-Szekeres Theorem proved by Fishburn and Graham $30$ years ago. Their bound was recently improved by Bucić, Sudakov, and Tran, who gave an upper bound that is triply exponential in four or more dimensions. We improve upon their results showing that a doubly expoenential upper bounds holds any number of dimensions.

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arXiv2024

Monochromatic odd cycles in edge-coloured complete graphs

It is easy to see that every $q$-edge-colouring of the complete graph on $2^q+1$ vertices must contain a monochromatic odd cycle. A natural question raised by Erdős and Graham in $1973$ asks for the smallest $L(q)$ such that every $q$-edge-colouring of $K_{2^q+1}$ must contain a monochromatic odd cycle of length at most $L(q)$. In here, we show that $L(q)=O\left(\frac{2^q}{q^{1-o(1)}}\right)$ giving the first non-trivial upper bound on $L(q)$.

math.CO↗

arXiv2024

Sparse Partitions of Graphs with Bounded Clique Number

We prove that for each integer $r\geq 2$, there exists a constant $C_r>0$ with the following property: for any $0<\varepsilon \leq 1/2$ and any graph $G$ with clique number at most $r,$ there is a partition of $V(G)$ into at most $(1/\varepsilon)^{C_r}$ sets $S_1, \dots, S_t,$ such that $G[S_i]$ has maximum degree at most $\varepsilon |S_i|$ for each $1 \leq i \leq t.$ This answers a question of Fox, Nguyen, Scott and Seymour, who proved a similar result for graphs with no induced $P_4.$

math.CO↗

arXiv2024

Immersions of directed graphs in tournaments

Recently, Draganić, Munhá Correia, Sudakov and Yuster showed that every tournament on $(2+o(1))k^2$ vertices contains a $1$-subdivision of a transitive tournament on $k$ vertices, which is tight up to a constant factor. We prove a counterpart of their result for immersions. Let $f(k)$ be the smallest integer such that any tournament on at least $f(k)$ vertices must contain a $1$-immersion of a transitive tournament on $k$ vertices. We show that $f(k)=O(k)$, which is clearly tight up to a multiplicative factor. If one insists in finding an immersion of a complete directed graph on $k$ vertices then an extra condition on the tournament is necessary. Indeed, we show that every tournament with minimum out-degree at least $Ck$ must contain a $2$-immersion of a complete digraph on $k$ vertices. This is again tight up to the value of $C$ and tight on the order of the paths in the immersion.

math.CO↗

arXiv2024

Seymour's second neighbourhood conjecture: random graphs and reductions

A longstanding conjecture of Seymour states that in every oriented graph there is a vertex whose second outneighbourhood is at least as large as its outneighbourhood. In this short note we show that, for any fixed $p\in[0,1/2)$, a.a.s. every orientation of $G(n,p)$ satisfies Seymour's conjecture (as well as a related conjecture of Sullivan). This improves on a recent result of Botler, Moura and Naia. Moreover, we show that $p=1/2$ is a natural barrier for this problem, in the following sense: for any fixed $p\in(1/2,1)$, Seymour's conjecture is actually equivalent to saying that, with probability bounded away from $0$, every orientation of $G(n,p)$ satisfies Seymour's conjecture. This provides a first reduction of the problem. For a second reduction, we consider minimum degrees and show that, if Seymour's conjecture is false, then there must exist arbitrarily large strongly-connected counterexamples with bounded minimum outdegree. Contrasting this, we show that vertex-minimal counterexamples must have large minimum outdegree.

math.CO↗

arXiv2024

Small families of partially shattering permutations

We say that a family of permutations $t$-shatters a set if it induces at least $t$ distinct permutations on that set. What is the minimum number $f_k(n,t)$ of permutations of $\{1, \dots, n\}$ that $t$-shatter all subsets of size $k$? For $t \le 2$, $f_k(n,t) = Θ(1)$. Spencer showed that $f_k(n,t) = Θ(\log \log n)$ for $3 \le t \le k$ and $f_k(n,k!) = Θ(\log n)$. In 1996, Füredi asked whether partial shattering with permutations must always fall into one of these three regimes. Johnson and Wickes recently settled the case $k = 3$ affirmatively and proved that $f_k(n,t) = Θ(\log n)$ for $t > 2 (k-1)!$. We give a surprising negative answer to the question of Füredi by showing that a fourth regime exists for $k \ge 4$. We establish that $f_k(n,t) = Θ(\sqrt{\log n})$ for certain values of $t$ and prove that this is the only other regime when $k = 4$. We also show that $f_k(n,t) = Θ(\log n)$ for $t > 2^{k-1}$. This greatly narrows the range of $t$ for which the asymptotic behaviour of $f_k(n,t)$ is unknown.

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