Searcharxiv⌕ Search

arXiv subjects

Michael Krivelevich

Publications and source records attributed to Michael Krivelevich.

At least 73 records · Page 4Linked to original sources
arXiv2021

Discrepancies of Spanning Trees and Hamilton Cycles

We study the multicolour discrepancy of spanning trees and Hamilton cycles in graphs. As our main result, we show that under very mild conditions, the $r$-colour spanning-tree discrepancy of a graph $G$ is equal, up to a constant, to the minimum $s$ such that $G$ can be separated into $r$ equal parts by deleting $s$ vertices. This result arguably resolves the question of estimating the spanning-tree discrepancy in essentially all graphs of interest. In particular, it allows us to immediately deduce as corollaries most of the results that appear in a recent paper of Balogh, Csaba, Jing and Pluhár, proving them in wider generality and for any number of colours. We also obtain several new results, such as determining the spanning-tree discrepancy of the hypercube. For the special case of graphs possessing certain expansion properties, we obtain exact asymptotic bounds. We also study the multicolour discrepancy of Hamilton cycles in graphs of large minimum degree, showing that in any $r$-colouring of the edges of a graph with $n$ vertices and minimum degree at least $\frac{r+1}{2r}n + d$, there must exist a Hamilton cycle with at least $\frac{n}{r} + 2d$ edges in some colour. This extends a result of Balogh et al., who established the case $r = 2$. The constant $\frac{r+1}{2r}$ in this result is optimal; it cannot be replaced by any smaller constant.

math.CO↗
arXiv2021

Expansion, long cycles, and complete minors in supercritical random subgraphs of the hypercube

Analogous to the case of the binomial random graph $G(d+1,p)$, it is known that the behaviour of a random subgraph of a $d$-dimensional hypercube, where we include each edge independently with probability $p$, which we denote by $Q^d_p$, undergoes a phase transition around the critical value of $p=\frac{1}{d}$. More precisely, standard arguments show that significantly below this value of $p$, with probability tending to one as $d \to \infty$ (whp for short) all components of this graph have order $O(d)$, whereas Ajtai, Komlós and Szemerédi showed that significantly above this value, in the \emph{supercritical regime}, whp there is a unique `giant' component of order $Θ\left(2^d\right)$. In $G(d+1,p)$ much more is known about the complex structure of the random graph which emerges in this supercritical regime. For example, it is known that in this regime whp $G(d+1,p)$ contains paths and cycles of length $Ω(d)$, as well as complete minors of order $Ω\left(\sqrt{d}\right)$. In this paper we obtain analogous results in $Q^d_p$. In particular, we show that for supercritical $p$, i.e., when $p=\frac{1+ε}{d}$ for a positive constant $ε$, whp $Q^d_p$ contains a cycle of length $Ω\left(\frac{2^d}{d^3(\log d)^3} \right)$ and a complete minor of order $Ω\left(\frac{2^{\frac{d}{2}}}{d^3(\log d)^3 }\right)$. In order to prove these results, we show that whp the largest component of $Q^d_p$ has good edge-expansion properties, a result of independent interest. We also consider the genus of $Q^d_p$ and show that, in this regime of $p$, whp the genus is $Ω\left(2^d\right)$.

math.CO↗
arXiv2021

Expansion in supercritical random subgraphs of the hypercube and its consequences

It is well-known that the behaviour of a random subgraph of a $d$-dimensional hypercube, where we include each edge independently with probability $p$, undergoes a phase transition when $p$ is around $\frac{1}{d}$. More precisely, standard arguments show that just below this value of $p$ all components of this graph have order $O(d)$ with probability tending to one as $d \to \infty$ (whp for short), whereas Ajtai, Komlós and Szemerédi [Largest random component of a $k$-cube, Combinatorica 2 (1982), no. 1, 1--7; MR0671140] showed that just above this value, in the supercritical regime, whp there is a unique `giant' component of order $Θ\left(2^d\right)$. We show that whp the vertex-expansion of the giant component is inverse polynomial in $d$. As a consequence we obtain polynomial in $d$ bounds on the diameter of the giant component and the mixing time of the lazy random walk on the giant component, answering questions of Bollobás, Kohayakawa and Łuczak [On the diameter and radius of random subgraphs of the cube, Random Structures and Algorithms 5 (1994), no. 5, 627--648; MR1300592] and of Pete [A note on percolation on $\mathbb{Z}^d$: isoperimetric profile via exponential cluster repulsion, Electron. Commun. Probab. 13 (2008), 377--392; MR2415145]. Furthermore, our results imply lower bounds on the circumference and Hadwiger number of a random subgraph of the hypercube in this regime of $p$ which are tight up to polynomial factors in $d$.

math.CO↗
arXiv2021

On subgraphs with degrees of prescribed residues in the random graph

We show that with high probability the random graph $G_{n, 1/2}$ has an induced subgraph of linear size, all of whose degrees are congruent to $r\pmod q$ for any fixed $r$ and $q\geq 2$. More generally, the same is true for any fixed distribution of degrees modulo $q$. Finally, we show that with high probability we can partition the vertices of $G_{n, 1/2}$ into $q+1$ parts of nearly equal size, each of which induces a subgraph all of whose degrees are congruent to $r\pmod q$. Our results resolve affirmatively a conjecture of Scott, who addressed the case $q=2$.

math.CO↗
arXiv2021

Hitting time of edge disjoint Hamilton cycles in random subgraph processes on dense base graphs

Consider the random subgraph process on a base graph $G$ on $n$ vertices: a sequence $\lbrace G_t \rbrace _{t=0} ^{|E(G)|}$ of random subgraphs of $G$ obtained by choosing an ordering of the edges of $G$ uniformly at random, and by sequentially adding edges to $G_0$, the empty graph on the vertex set of $G$, according to the chosen ordering. We show that if $G$ has one of the following properties: 1. There is a positive constant $\varepsilon > 0$ such that $δ(G) \geq \left( \frac{1}{2} + \varepsilon \right) n$; 2. There are some constants $α, β>0$ such that every two disjoint subsets $U,W$ of size at least $αn$ have at least $β|U||W|$ edges between them, and the minimum degree of $G$ is at least $(2α+ β)\cdot n$; or: 3. $G$ is an $(n,d,λ)$--graph, with $d\geq \frac{C\cdot n\cdot \log \log n}{\log n}$ and $λ\leq \frac{c\cdot d^2}{n}$ for some absolute constants $c,C>0$. then for a positive integer constant $k$ with high probability the hitting time of the property of containing $k$ edge disjoint Hamilton cycles is equal to the hitting time of having minimum degree at least $2k$. These results extend prior results by by Johansson and by Frieze and Krivelevich, and answer a question posed by Frieze.

math.CO↗
arXiv2021

Divisible subdivisions

We prove that for every graph $H$ of maximum degree at most $3$ and for every positive integer $q$ there is a finite $f=f(H,q)$ such that every $K_f$-minor contains a subdivision of $H$ in which every edge is replaced by a path whose length is divisible by $q$. For the case of cycles we show that for $f=O(q \log q)$ every $K_f$-minor contains a cycle of length divisible by $q$, and observe that this settles a recent problem of Friedman and the second author about cycles in (weakly) expanding graphs.

math.CO↗
arXiv2021

Colour-biased Hamilton cycles in random graphs

We prove that a random graph $G(n,p)$, with $p$ above the Hamiltonicity threshold, is typically such that for any $r$-colouring of its edges there exists a Hamilton cycle with at least $(2/(r+ 1)-o(1))n$ edges of the same colour. This estimate is asymptotically optimal.

math.CO↗
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).

math.CO↗
arXiv2020

Component Games on Random Graphs

In the $\left(1:b\right)$ component game played on a graph $G$, two players, Maker and Breaker, alternately claim~$1$ and~$b$ previously unclaimed edges of $G$, respectively. Maker's aim is to maximise the size of a largest connected component in her graph, while Breaker is trying to minimise it. We show that the outcome of the game on the binomial random graph is strongly correlated with the appearance of a nonempty $(b+2)$-core in the graph. For any integer $k$, the $k$-core of a graph is its largest subgraph of minimum degree at least $k$. Pittel, Spencer and Wormald showed in 1996 that for any $k\ge3$ there exists an explicitly defined constant $c_{k}$ such that $p=c_{k}/n$ is the threshold function for the appearance of the $k$-core in $G(n,p)$. More precisely, $G(n,c/n)$ has WHP a linear-size $k$-core when the constant $c>c_{k}$, and an empty $k$-core when $c c_{b+2}$, while Breaker can WHP prevent Maker from building larger than polylogarithmic-size components if $c<c_{b+2}$. For Breaker's strategy, we prove a theorem which may be of independent interest. The standard algorithm for computing the $k$-core of any graph is to repeatedly delete ("peel") all vertices of degree less than $k$, as long as such vertices remain. When $G(n,c/n)$ for $c<c_{k}$, it was shown by Jiang, Mitzenmacher and Thaler that $\log_{k-1}\log n+Θ(1)$ peeling iterations are WHP necessary and sufficient to obtain the (empty) $k$-core of~$G$. Our theorem states that already after a constant number of iterations, $G$ is WHP shattered into pieces of polylogarithmic size.

math.CO↗
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 $H$.

math.CO↗
arXiv2020

Large complete minors in random subgraphs

Let $G$ be a graph of minimum degree at least $k$ and let $G_p$ be the random subgraph of $G$ obtained by keeping each edge independently with probability $p$. We are interested in the size of the largest complete minor that $G_p$ contains when $p = \frac{1+\varepsilon}{k}$ with $\varepsilon >0$. We show that with high probability $G_p$ contains a complete minor of order $\tildeΩ(\sqrt{k})$, where the $\sim$ hides a polylogarithmic factor. Furthermore, in the case where the order of $G$ is also bounded above by a constant multiple of $k$, we show that this polylogarithmic term can be removed, giving a tight bound.

math.CO↗
arXiv2020

Very fast construction of bounded-degree spanning graphs via the semi-random graph process

Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some predetermined objective in an online randomized environment. They have algorithmic implications in various areas of computer science, as well as connections to biological processes involving decision making. In this paper, we consider a recently proposed semi-random graph process, described as follows: we start with an empty graph on $n$ vertices, and in each round, the decision-maker, called Builder, receives a uniformly random vertex $v$, and must immediately (in an online manner) choose another vertex $u$, adding the edge $\{u,v\}$ to the graph. Builder's end goal is to make the constructed graph satisfy some predetermined monotone graph property. We consider the property of containing a spanning graph $H$ as a subgraph. It was asked by N. Alon whether for any bounded-degree $H$, Builder can construct a copy of $H$ w.h.p. in $O(n)$ rounds. We answer this question positively in a strong sense, showing that any graph with maximum degree $Δ$ can be constructed w.h.p. in $(3Δ/2 + o(Δ)) n$ rounds. This is tight (even for the offline case) up to a multiplicative factor of $3 + o_Δ(1)$. Furthermore, for the special case where $H$ is a spanning forest of maximum degree $Δ$, we show that $H$ can be constructed w.h.p. in $O(n \log Δ)$ rounds. This is tight up to a multiplicative constant, even for the offline setting. Finally, we show a separation between adaptive and non-adaptive strategies, proving a lower bound of $Ω(n\sqrt{\log n})$ on the number of rounds necessary to eliminate all isolated vertices w.h.p. using a non-adaptive strategy. This bound is tight, and in fact there are non-adaptive strategies for constructing a Hamilton cycle or a $K_r$-factor, which are successful w.h.p. within $O(n\sqrt{\log n})$ rounds.

math.CO↗
arXiv2020

Cycle lengths in sparse random graphs

We study the set ${\cal L}(G)$ of lengths of all cycles that appear in a random $d$-regular $G$ on $n$ vertices for a fixed $d\geq 3$, as well as in Erdős--Rényi random graphs on $n$ vertices with a fixed average degree $c>1$. Fundamental results on the distribution of cycle counts in these models were established in the 1980's and early 1990's, with a focus on the extreme lengths: cycles of fixed length, and cycles of length linear in $n$. Here we derive, for a random $d$-regular graph, the limiting probability that ${\cal L}(G)$ simultaneously contains the entire range $\{\ell,\ldots,n\}$ for $\ell\geq 3$, as an explicit expression $θ_\ell=θ_\ell(d)\in(0,1)$ which goes to $1$ as $\ell\to\infty$. For the random graph ${\cal G}(n,p)$ with $p=c/n$, where $c\geq C_0$ for some absolute constant $C_0$, we show the analogous result for the range $\{\ell,\ldots,(1-o(1))L_{\max}(G)\}$, where $L_{\max}$ is the length of a longest cycle in $G$. The limiting probability for ${\cal G}(n,p)$ coincides with $θ_\ell$ from the $d$-regular case when $c$ is the integer $d-1$. In addition, for the directed random graph ${\cal D}(n,p)$ we show results analogous to those on ${\cal G}(n,p)$, and for both models we find an interval of $c ε^2 n$ consecutive cycle lengths in the slightly supercritical regime $p=\frac{1+ε}n$.

math.CO↗
arXiv2020

Turán-type problems for long cycles in random and pseudo-random graphs

We study the Turán number of long cycles in random graphs and in pseudo-random graphs. Denote by $ex(G(n,p),H)$ the random variable counting the number of edges in a largest subgraph of $G(n,p)$ without a copy of $H$. We determine the asymptotic value of $ex(G(n,p), C_t)$ where $C_t$ is a cycle of length $t$, for $p\geq \frac Cn$ and $A \log n \leq t \leq (1 - \varepsilon)n$. The typical behavior of $ex(G(n,p), C_t)$ depends substantially on the parity of $t$. In particular, our results match the classical result of Woodall on the Turán number of long cycles, and can be seen as its random version, showing that the transference principle holds here as well. In fact, our techniques apply in a more general sparse pseudo-random setting. We also prove a robustness-type result, showing the likely existence of cycles of prescribed lengths in a random subgraph of a graph with a nearly optimal density.

math.CO↗
arXiv2020

The Kőnig Graph Process

Say that a graph G has property $\mathcal{K}$ if the size of its maximum matching is equal to the order of a minimal vertex cover. We study the following process. Set $N:= \binom{n}{2}$ and let $e_1, e_2, \dots e_{N}$ be a uniformly random ordering of the edges of $K_n$, with $n$ an even integer. Let $G_0$ be the empty graph on $n$ vertices. For $m \geq 0$, $G_{m+1}$ is obtained from $G_m$ by adding the edge $e_{m+1}$ exactly if $G_m \cup \{ e_{m+1}\}$ has property $\mathcal{K}$. We analyse the behaviour of this process, focusing mainly on two questions: What can be said about the structure of $G_N$ and for which $m$ will $G_m$ contain a perfect matching?

math.CO↗
arXiv2020

Cycle lengths in expanding graphs

For a positive constant $α$ a graph $G$ on $n$ vertices is called an $α$-expander if every vertex set $U$ of size at most $n/2$ has an external neighborhood whose size is at least $α\left|U\right|$. We study cycle lengths in expanding graphs. We first prove that cycle lengths in $α$-expanders are well distributed. Specifically, we show that for every $0<α\leq1$ there exist positive constants $n_{0}$, $C$ and $A=O(1/α)$ such that for every $α$-expander $G$ on $n\geq n_{0}$ vertices and every integer $\ell\in\left[C\log n,\frac{n}{C}\right]$, $G$ contains a cycle whose length is between $\ell$ and $\ell+A$; the order of dependence of the additive error term $A$ on $α$ is optimal. Secondly, we show that every $α$-expander on $n$ vertices contains $Ω\left(\frac{α^{3}}{\log\left(1/α\right)}\right)n$ different cycle lengths. Finally, we introduce another expansion-type property, guaranteeing the existence of a linearly long interval in the set of cycle lengths. For $β>0$ a graph $G$ on $n$ vertices is called a $β$-graph if every pair of disjoint sets of size at least $βn$ are connected by an edge. We prove that for every $β<1/20$ there exist positive constants $b_{1}=O\left(\frac{1}{\log\left(1/β\right)}\right)$ and $b_{2}=O\left(β\right)$ such that every $β$-graph $G$ on $n$ vertices contains a cycle of length $\ell$ for every integer $\ell\in\left[b_{1}\log n,(1-b_{2})n\right]$; the order of dependence of $b_{1}$ and $b_{2}$ on $β$ is optimal.

math.CO↗