arXiv subjects

Nemanja Draganić

Publications and source records attributed to Nemanja Draganić.

35 records · Page 2Linked to original sources

arXiv2023

Cycles with many chords

How many edges in an $n$-vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erdős and Staton considered this question and showed that any $n$-vertex graph with $2n^{3/2}$ edges contains such a cycle. We significantly improve this old bound by showing that $Ω(n\log^8n)$ edges are enough to guarantee the existence of such a cycle. Our proof exploits a delicate interplay between certain properties of random walks in almost regular expanders. We argue that while the probability that a random walk of certain length in an almost regular expander is self-avoiding is very small, one can still guarantee that it spans many edges (and that it can be closed into a cycle) with large enough probability to ensure that these two events happen simultaneously.

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arXiv2023

Effective bounds for induced size-Ramsey numbers of cycles

The induced size-Ramsey number $\hat{r}_\text{ind}^k(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have such that for any $k$-coloring of its edges, there exists a monochromatic copy of $H$ which is an induced subgraph of $G$. In 1995, in their seminal paper, Haxell, Kohayakawa and Luczak showed that for cycles, these numbers are linear for any constant number of colours, i.e., $\hat{r}_\text{ind}^k(C_n)\leq Cn$ for some $C=C(k)$. The constant $C$ comes from the use of the regularity lemma, and has a tower type dependence on $k$. In this paper we significantly improve these bounds, showing that $\hat{r}_\text{ind}^k(C_n)\leq O(k^{102})n$ when $n$ is even, thus obtaining only a polynomial dependence of $C$ on $k$. We also prove $\hat{r}_\text{ind}^k(C_n)\leq e^{O(k\log k)}n$ for odd $n$, which almost matches the lower bound of $e^{Ω(k)}n$. Finally, we show that the ordinary (non-induced) size-Ramsey number satisfies $\hat{r}^k(C_n)=e^{O(k)}n$ for odd $n$. This substantially improves the best previous result of $e^{O(k^2)}n$, and is best possible, up to the implied constant in the exponent. To achieve our results, we present a new host graph construction which, roughly speaking, reduces our task to finding a cycle of approximate given length in a graph with local sparsity.

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arXiv2023

A generalization of Bondy's pancyclicity theorem

The bipartite independence number of a graph $G$, denoted as $\tildeα(G)$, is the minimal number $k$ such that there exist positive integers $a$ and $b$ with $a+b=k+1$ with the property that for any two sets $A,B\subseteq V(G)$ with $|A|=a$ and $|B|=b$, there is an edge between $A$ and $B$. McDiarmid and Yolov showed that if $δ(G)\geq\tilde α(G)$ then $G$ is Hamiltonian, extending the famous theorem of Dirac which states that if $δ(G)\geq |G|/2$ then $G$ is Hamiltonian. In 1973, Bondy showed that, unless $G$ is a complete bipartite graph, Dirac's Hamiltonicity condition also implies pancyclicity, i.e., existence of cycles of all the lengths from $3$ up to $n$. In this paper we show that $δ(G)\geq\tilde α(G)$ implies that $G$ is pancyclic or that $G=K_{\frac{n}{2},\frac{n}{2}}$, thus extending the result of McDiarmid and Yolov, and generalizing the classic theorem of Bondy.

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arXiv2023

Chvátal-Erdős condition for pancyclicity

An $n$-vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices and it is pancyclic if it contains cycles of all lengths from $3$ up to $n$. A celebrated meta-conjecture of Bondy states that every non-trivial condition implying Hamiltonicity also implies pancyclicity (up to possibly a few exceptional graphs). We show that every graph $G$ with $κ(G) > (1+o(1)) α(G)$ is pancyclic. This extends the famous Chvátal-Erdős condition for Hamiltonicity and proves asymptotically a $30$-year old conjecture of Jackson and Ordaz.

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arXiv2022

Short proofs for long induced paths

We present a modification of the Depth first search algorithm, suited for finding long induced paths. We use it to give simple proofs of the following results. We show that the induced size-Ramsey number of paths satisfies $\hat{R}_{\mathrm{ind}}(P_n)\leq 5\cdot 10^7n$, thus giving an explicit constant in the linear bound, improving the previous bound with a large constant from a regularity lemma argument by Haxell, Kohayakawa and Łuczak. We also provide a bound for the $k$-color version, showing that $\hat{R}_{\mathrm{ind}}^k(P_n)=O(k^3\log^4k)n$. Finally, we present a new short proof of the fact that the binomial random graph in the supercritical regime, $G(n,\frac{1+\varepsilon}{n})$, contains typically an induced path of length $Θ(\varepsilon^2) n$.

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arXiv2022

The largest hole in sparse random graphs

We show that for any $d=d(n)$ with $d_0(ε) \le d =o(n)$, with high probability, the size of a largest induced cycle in the random graph $G(n,d/n)$ is $(2\pm ε)\frac{n}{d}\log d$. This settles a long-standing open problem in random graph theory.

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arXiv2021

Ramsey number of 1-subdivisions of transitive tournaments

The study of problems concerning subdivisions of graphs has a rich history in extremal combinatorics. Confirming a conjecture of Burr and Erdős, Alon proved in 1994 that subdivided graphs have linear Ramsey numbers. Later, Alon, Krivelevich and Sudakov showed that every $n$-vertex graph with at least $\varepsilon n^2$ edges contains a $1$-subdivision of the complete graph on $c_{\varepsilon}\sqrt{n}$ vertices, resolving another old conjecture of Erdős. In this paper we consider the directed analogue of these problems and show that every tournament on at least $(2+o(1))k^2$ vertices contains the 1-subdivision of a transitive tournament on $k$ vertices. This is optimal up to a multiplicative factor of 4 and confirms a conjecture of Girão, Popielarz and Snyder.

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arXiv2021

Tight bounds for divisible subdivisions

Alon and Krivelevich proved that for every $n$-vertex subcubic graph $H$ and every integer $q \ge 2$ there exists a (smallest) integer $f=f(H,q)$ such that every $K_f$-minor contains a subdivision of $H$ in which the length of every subdivision-path is divisible by $q$. Improving their superexponential bound, we show that $f(H,q) \le \frac{21}{2}qn+8n+14q$, which is optimal up to a constant multiplicative factor.

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arXiv2021

Tight bound for powers of Hamilton cycles in tournaments

A basic result in graph theory says that any $n$-vertex tournament with in- and out-degrees larger than $\frac{n-2}{4}$ contains a Hamilton cycle, and this is tight. In 1990, Bollobás and Häggkvist significantly extended this by showing that for any fixed $k$ and $\varepsilon > 0$, and sufficiently large $n$, all tournaments with degrees at least $\frac{n}{4}+\varepsilon n$ contain the $k$-th power of a Hamilton cycle. Up until now, there has not been any progress on determining a more accurate error term in the degree condition, neither in understanding how large $n$ should be in the Bollobás-Häggkvist theorem. We essentially resolve both of these questions. First, we show that if the degrees are at least $\frac{n}{4} + cn^{1-1/\lceil k/2 \rceil}$ for some constant $c = c(k)$, then the tournament contains the $k$-th power of a Hamilton cycle. In particular, in order to guarantee the square of a Hamilton cycle, one only requires a constant additive term. We also present a construction which, modulo a well-known conjecture on Turán numbers for complete bipartite graphs, shows that the error term must be of order at least $n^{1-1/\lceil (k-1)/2 \rceil}$, which matches our upper bound for all even $k$. For odd $k$, we believe that the lower bound can be improved. Indeed, we show that for $k=3$, there exist tournaments with degrees $\frac{n}{4}+Ω(n^{1/5})$ and no cube of a Hamilton cycle. In addition, our results imply that the Bollobás-Häggkvist theorem already holds for $n = \varepsilon^{-Θ(k)}$, which is best possible.

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arXiv2021

Rolling backwards can move you forward: on embedding problems in sparse expanders

We develop a general embedding method based on the Friedman-Pippenger tree embedding technique (1987) and its algorithmic version, essentially due to Aggarwal et al. (1996), enhanced with a roll-back idea allowing to sequentially retrace previously performed embedding steps. We use this method to obtain the following results. -We show that the size-Ramsey number of logarithmically long subdivisions of bounded degree graphs is linear in their number of vertices, settling a conjecture of Pak (2002). -We give a deterministic, polynomial time online algorithm for finding vertex-disjoint paths of prescribed length between given pairs of vertices in an expander graph. Our result answers a question of Alon and Capalbo (2007). -We show that relatively weak bounds on the spectral ratio of $d$-regular graphs force the existence of a topological minor of $K_t$ where $t=(1-o(1))d$. We also exhibit a construction which shows that the theoretical maximum $t=d+1$ cannot be attained even if $λ=O(\sqrt{d})$. This answers a question of Fountoulakis, Kühn and Osthus (2009).

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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

Universal and unavoidable graphs

The Turán number $\text{ex}(n,H)$ of a graph $H$ is the maximal number of edges in an $H$-free graph on $n$ vertices. In $1983$ Chung and Erdős asked which graphs $H$ with $e$ edges minimize $\text{ex}(n,H)$. They resolved this question asymptotically for most of the range of $e$ and asked to complete the picture. In this paper we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting we extend previous work done by Babai, Chung, Erdős, Graham and Spencer, and by Alon and Asodi.

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arXiv2020

Large induced matchings in random graphs

Given a large graph $H$, does the binomial random graph $G(n,p)$ contain a copy of $H$ as an induced subgraph with high probability? This classical question has been studied extensively for various graphs $H$, going back to the study of the independence number of $G(n,p)$ by Erdős and Bollobás, and Matula in 1976. In this paper we prove an asymptotically best possible result for induced matchings by showing that if $C/n\le p \le 0.99$ for some large constant $C$, then $G(n,p)$ contains an induced matching of order approximately $2\log_q(np)$, where $q= \frac{1}{1-p}$.

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arXiv2020

The size-Ramsey number of short subdivisions

The $r$-size-Ramsey number $\hat{R}_r(H)$ of a graph $H$ is the smallest number of edges a graph $G$ can have, such that for every edge-coloring of $G$ with $r$ colors there exists a monochromatic copy of $H$ in $G$. For a graph $H$, we denote by $H^q$ the graph obtained from $H$ by subdividing its edges with $q{-}1$ vertices each. In a recent paper of Kohayakawa, Retter and R{ö}dl, it is shown that for all constant integers $q,r\geq 2$ and every graph $H$ on $n$ vertices and of bounded maximum degree, the $r$-size-Ramsey number of $H^q$ is at most $(\log n)^{20(q-1)}n^{1+1/q}$, for $n$ large enough. We improve upon this result using a significantly shorter argument by showing that $\hat{R}_r(H^q)\leq O(n^{1+1/q})$ for any such graph $H$.

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arXiv2020

A Note on Powers of Paths in Tournaments

In this note we show that every tournament on $n$ vertices contains the $k$-th power of a directed path of length $n/2^{6k+7}$, which improves upon the recent bound of Scott and Korándi of $n/2^{2^{3k}}$. By doing so, we get an inverse exponential dependence on $k$, which is best possible as Yuster recently showed an upper bound of $kn/{2^{k/2}}$.

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arXiv2020

Large induced trees in dense random graphs

Erdős and Palka initiated the study of the maximal size of induced trees in random graphs in 1983. They proved that for every fixed $0<p<1$ the size of a largest induced tree in $G_{n,p}$ is concentrated around $2\log_q (np)$ with high probability, where $q=(1-p)^{-1}$. De la Vega showed concentration around the same value for $p=C/n$ where $C$ is a large constant, and his proof also works for all larger $p$. We show that for any given tree $T$ with bounded maximum degree and of size $(2-o(1))\log_q(np)$, $G_{n,p}$ contains an induced copy of $T$ with high probability for $n^{-1/2}\ln^{10/9}n\leq p\leq 0.99$. This is asymptotically optimal.

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arXiv2019

A Ramsey Theorem for Multiposets

In the parlance of relational structures, the Finite Ramsey Theorem states that the class of all finite chains has the Ramsey property. A classical result of J. Nešetřil and V. Rödl claims that the class of all finite posets with a linear extension has the Ramsey property. In 2010 M. Sokić proved that the class of all finite structures consisting of several linear orders has the Ramsey property. This was followed by a 2017 result of S. Solecki and M. Zhao that the class of all finite posets with several linear extensions has the Ramsey property. Using the categorical reinterpretation of the Ramsey property in this paper we prove a common generalization of all these results. We consider multiposets to be structures consisting of several partial orders and several linear orders. We allow partial orders to extend each other in an arbitrary but fixed way, and require that every partial order is extended by at least one of the linear orders. We then show that the class of all finite multiposets conforming to a fixed template has the Ramsey property.

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