Guilherme Oliveira Mota's research while affiliated with University of São Paulo and other places
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Publications (23)
For every k⩾3$$ k\geqslant 3 $$, we determine the order of growth, up to polylogarithmic factors, of the number of orientations of the binomial random graph containing no directed cycle of length k$$ k $$. This solves a conjecture of Kohayakawa, Morris and the last two authors.
A graph is locally irregular if any pair of adjacent vertices have distinct degrees. A locally irregular decomposition of a graph G is a decomposition of G into subgraphs that are locally irregular. We prove that any graph G can be decomposed into at most 2∆(G) − 1 locally irregular graphs, improving on the previous upper bound of 3∆(G)−2. We also...
Seja G = G(n, p) o grafo aleatório binomial. Provamos que se p » (ln n/n)1/2, então com alta probabilidade toda aresta-coloração própria de G admite uma cobertura de E(G) por O(n) caminhos multicoloridos, em que uma cópia de um grafo é dita multicolorida se todas as suas arestas possuem cores distintas.
The celebrated canonical Ramsey theorem of Erd\H{o}s and Rado implies that for a given graph $H$, if $n$ is sufficiently large then any colouring of the edges of $K_n$ gives rise to copies of $H$ that exhibit certain colour patterns, namely monochromatic, rainbow or lexicographic. We are interested in sparse random versions of this result and the t...
Let Sk(n) be the maximum number of orientations of an n-vertex graph G in which no copy of Kk is strongly connected. For all integers n, k≥4 where n≥5 or k≥5, we prove that Sk(n)=2tk−1(n), where tk−1(n) is the number of edges of the n-vertex (k−1)-partite Turán graph Tk−1(n), and that Tk−1(n) is the only n-vertex graph with this number of orientati...
We investigate the threshold $p_{\vec H}=p_{\vec H}(n)$ for the Ramsey-type property $G(n,p)\to \vec H$, where $G(n,p)$ is the binomial random graph and $G\to\vec H$ indicates that every orientation of the graph $G$ contains the oriented graph $\vec H$ as a subdigraph. Similarly to the classical Ramsey setting, the upper bound $p_{\vec H}\leq Cn^{-...
For every $k \geq 3$, we determine the order of growth, up to polylogarithmic factors, of the number of orientations of the binomial random graph containing no directed cycle of length $k$. This solves a conjecture of Kohayakawa, Morris and the last two authors.
In this paper, we consider the problem of computing the minimum number of colors needed to properly color the edges of a complete graph on $n$ vertices so that there are no pair of vertex-disjoint triangles colored with the same colors. This problem was introduced recently (in a more general context) by Conlon and Tyomkyn, and the corresponding val...
Given graphs $G$, $H_1$, and $H_2$, let $G\xrightarrow{\text{mr}}(H_1,H_2)$ denote the property that in every edge colouring of $G$ there is a monochromatic copy of $H_1$ or a rainbow copy of $H_2$. The constrained Ramsey number, defined as the minimum $n$ such that $K_n\xrightarrow{\text{mr}}(H_1,H_2)$, exists if and only if $H_1$ is a star or $H_...
For graphs $G$ and $H$, we write $G \overset{\mathrm{rb}}{\longrightarrow} H $ if any proper edge-coloring of $G$ contains a rainbow copy of $H$, i.e., a copy where no color appears more than once. Kohayakawa, Konstadinidis and the last author proved that the threshold for $G(n,p) \overset{\mathrm{rb}}{\longrightarrow}H$ is at most $n^{-1/m_2(H)}$....
For graphs G and H, let G⟶rbH denote the property that for every proper edge colouring of G there is a rainbow copy of H in G. Extending a result of Nenadov et al. (2017), we determine the threshold for G(n,p)⟶rbCℓ for cycles Cℓ of any given length ℓ≥4.
We determine the order of growth, up to polylogarithmic factors, of the number of orientations of the binomial random graph G(n, p) containing no directed cycle of length k for every k⩾3. This solves a conjecture of Kohayakawa, Morris and the last two authors.
For a graph G and an oriented graph H→, let G→H→ denote the property that every orientation of G contains a copy of H→. We investigate the threshold pH→=pH→(n) for G(n,p)→H→, where G(n, p) is the binomial random graph. Similarly to the classical (edge-colouring) Ramsey setting, pH→⩽n-1/m2(H→), where m2(H→) denotes the maximum 2-density of H→. While...
We determine, up to a multiplicative constant, the optimal number of random edges that need to be added to a k‐graph H with minimum vertex degree to ensure an F‐factor with high probability, for any F that belongs to a certain class of k‐graphs, which includes, for example, all k‐partite k‐graphs, and the Fano plane. In particular, taking F to be a...
Para grafos G, S e H, dizemos que G mr-flecha (S,H) se toda coloração das arestas de G tem uma cópia monocromática de S ou uma cópia multicolorida de H. Provamos que se S = K_{1,3} e H é uma árvore binária completa de altura h, então o tamanho da menor árvore T que mr-flecha(S,H) é 2^{(1/2+o(1))h^2}.
Given a hypergraph H, the size-Ramsey number r(H) is the smallest integer m such that there exists a graph G with m edges with the property that in any colouring of the edges of G with two colours there is amonochromatic copy of H. We prove that the size-Ramsey number of the 3-uniform tight path on n vertices P_n is linear in n, i.e., r(P_n)=O(n)....
Let $S_k(n)$ be the maximum number of orientations of an $n$-vertex graph $G$ in which no copy of $K_k$ is strongly connected. For all integers $n$, $k\geq 4$ where $n\geq 5$ or $k\geq 5$, we prove that $S_k(n) = 2^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is the number of edges of the $n$-vertex $(k-1)$-partite Tur\'an graph $T_{k-1}(n)$, and that $T_{k-1...
Given graphs G, H1 and H2, let G→mr(H1, H2) denote the property that in every edge-colouring of G there is a monochromatic copy of H1 or a rainbow copy of H2. The constrained Ramsey number, defined as the minimum n such that Kn→mr(H1, H2), exists if and only if H1 is a star or H2 is a forest. We determine the threshold for the property G(n,p) →mr(H...
Let Sk(n) be the maximum number of orientations of an n-vertex graph G in which no copy of Kk is strongly connected. For all integers n, k ≥ 4 where n ≥ 5 or k ≥ 5, we prove that Sk(n) = 2tk - 1(n), where tk-1(n) is the number of edges of the n-vertex (k - 1)-partite Turán graph Tk-1(n). Moreover, we prove that Tk-1(n) is the only graph having 2tk-...
Citations
... For example, λ " 1 corresponds to studying proper edge colourings of Gpn, pq and anti-Ramsey properties (see, e.g. [10,11,16] and the references therein). In fact, for ℓ ě 5, there are proper colourings of Gpn, pq with p " cn´2 ℓ`1 which do not contain a rainbow copy of K ℓ (see [11]), which is an alternative argument forp K ℓ ě cn´2 ℓ`1 and another obstruction for Theorem 1.3. ...
Reference: Canonical colourings in random graphs
![[object Object]](https://i1.rgstatic.net/ii/profile.image/589239930613771-1517497155964_Q64/Olaf-Parczyk.jpg)
... Bucić, Janzer and Sudakov [5] determined D(n, C ↻ 2 +1 ) for every ⩾ 1 as long as n is sufficiently large, extending the proof in [2]. Another extension of the results in [2] was given by Araújo, Botler, and the last author [3] who determined D(n, C ↻ 3 ) for every n ∈ N (see also [4]). ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/281169741533193-1444047498452_Q64/Fabio-Botler.jpg)
... The main results of this work were announced in the extended abstract [3]. ...
... Since the 2-color case was resolved completely by Kohayakawa, Mota, and Schacht, the 3-color case arises as the natural next step. This problem was recently considered by Kohayakawa, Mendonça, Mota, and Schülke [13] who proved that the threshold is at most (log n∕n) 1∕6 , improving the previous bound of (log n∕n) 1∕8 from [5]. In fact, (log n∕n) 1∕6 is also the hard limit of the approach of [5] and was implicitly conjectured there to be the answer. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/847264092258339-1579014909869_Q64/Bjarne-Schuelke.jpg)
... For k-graphs for k ≥ 3, Han et al. [16] proved a linear bound for 3-uniform tight paths, and Letzter, Pokrovskiy, Yepremyan [22] for all uniformities and more generally powers of bounded degree hypertrees. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/279603382898694-1443674049539_Q64/Jie-Han-6.jpg)
