Searcharxiv⌕ Search

arXiv subjects

Oleg Pikhurko

Publications and source records attributed to Oleg Pikhurko.

At least 73 records · Page 4Linked to original sources
arXiv2019

Minimizing the number of 5-cycles in graphs with given edge-density

Motivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of the 5-cycle $C_5$. We show that every graph of order $n$ and size $\left( 1-\frac{1}{k}\right)\binom{n}{2}$, where $k\ge 3$ is an integer, contains at least \[ \left( \frac{1}{10} -\frac{1}{2k} + \frac{1}{k^2} - \frac{1}{k^3} + \frac{2}{5 k^4} \right)n^5 +o(n^5) \] copies of $C_5$. This bound is optimal, since a matching upper bound is given by the balanced complete $k$-partite graph. The proof is based on the flag algebras framework. We also provide a stability result. An SDP solver is not necessary to verify our proofs.

math.CO↗
arXiv2019

Supersaturation Problem for the Bowtie

The Turán function $ex(n,F)$ denotes the maximal number of edges in an $F$-free graph on $n$ vertices. We consider the function $h_F(n,q)$, the minimal number of copies of $F$ in a graph on $n$ vertices with $ex(n,F)+q$ edges. The value of $h_F(n,q)$ has been extensively studied when $F$ is bipartite or colour-critical. In this paper we investigate the simplest remaining graph $F$, namely, two triangles sharing a vertex, and establish the asymptotic value of $h_F(n,q)$ for $q=o(n^2)$.

math.CO↗
arXiv2019

Minimum number of additive tuples in groups of prime order

For a prime number $p$ and a sequence of integers $a_0,\dots,a_k\in \{0,1,\dots,p\}$, let $s(a_0,\dots,a_k)$ be the minimum number of $(k+1)$-tuples $(x_0,\dots,x_k)\in A_0\times\dots\times A_k$ with $x_0=x_1+\dots + x_k$, over subsets $A_0,\dots,A_k\subseteq\mathbb{Z}_p$ of sizes $a_0,\dots,a_k$ respectively. An elegant argument of Lev (independently rediscovered by Samotij and Sudakov) shows that there exists an extremal configuration with all sets $A_i$ being intervals of appropriate length, and that the same conclusion also holds for the related problem, reposed by Bajnok, when $a_0=\dots=a_k=:a$ and $A_0=\dots=A_k$, provided $k$ is not equal 1 modulo $p$. By applying basic Fourier analysis, we show for Bajnok's problem that if $p\ge 13$ and $a\in\{3,\dots,p-3\}$ are fixed while $k\equiv 1\pmod p$ tends to infinity, then the extremal configuration alternates between at least two affine non-equivalent sets.

math.CO↗
arXiv2018

Edges not in any monochromatic copy of a fixed graph

For a sequence $(H_i)_{i=1}^k$ of graphs, let $\textrm{nim}(n;H_1,\ldots, H_k)$ denote the maximum number of edges not contained in any monochromatic copy of $H_i$ in colour $i$, for any colour $i$, over all $k$-edge-colourings of~$K_n$. When each $H_i$ is connected and non-bipartite, we introduce a variant of Ramsey number that determines the limit of $\textrm{nim}(n;H_1,\ldots, H_k)/{n\choose 2}$ as $n\to\infty$ and prove the corresponding stability result. Furthermore, if each $H_i$ is what we call \emph{homomorphism-critical} (in particular if each $H_i$ is a clique), then we determine $\textrm{nim}(n;H_1,\ldots, H_k)$ exactly for all sufficiently large~$n$. The special case $\textrm{nim}(n;K_3,K_3,K_3)$ of our result answers a question of Ma. For bipartite graphs, we mainly concentrate on the two-colour symmetric case (i.e., when $k=2$ and $H_1=H_2$). It is trivial to see that $\textrm{nim}(n;H,H)$ is at least $\textrm{ex}(n,H)$, the maximum size of an $H$-free graph on $n$ vertices. Keevash and Sudakov showed that equality holds if $H$ is the $4$-cycle and $n$ is large; recently Ma extended their result to an infinite family of bipartite graphs. We provide a larger family of bipartite graphs for which $\textrm{nim}(n;H,H)=\textrm{ex}(n,H)$. For a general bipartite graph $H$, we show that $\textrm{nim}(n;H,H)$ is always within a constant additive error from $\textrm{ex}(n,H)$, i.e.,~$\textrm{nim}(n;H,H)= \textrm{ex}(n,H)+O_H(1)$.

math.CO↗
arXiv2018

How unproportional must a graph be?

Let $u_k(G,p)$ be the maximum over all $k$-vertex graphs $F$ of by how much the number of induced copies of $F$ in $G$ differs from its expectation in the binomial random graph with the same number of vertices as $G$ and with edge probability $p$. This may be viewed as a measure of how close $G$ is to being $p$-quasirandom. For a positive integer $n$ and $0<p<1$, let $D(n,p)$ be the distance from $p\binom{n}{2}$ to the nearest integer. Our main result is that, for fixed $k\ge 4$ and for $n$ large, the minimum of $u_k(G,p)$ over $n$-vertex graphs has order of magnitude $Θ\big(\max\{D(n,p), p(1-p)\} n^{k-2}\big)$ provided that $p(1-p)n^{1/2} \to \infty$.

math.CO↗
arXiv2018

Strong Forms of Stability from Flag Algebra Calculations

Given a hereditary family $\mathcal{G}$ of admissible graphs and a function $λ(G)$ that linearly depends on the statistics of order-$κ$ subgraphs in a graph $G$, we consider the extremal problem of determining $λ(n,\mathcal{G})$, the maximum of $λ(G)$ over all admissible graphs $G$ of order $n$. We call the problem perfectly $B$-stable for a graph $B$ if there is a constant $C$ such that every admissible graph $G$ of order $n\ge C$ can be made into a blow-up of $B$ by changing at most $C(λ(n,\mathcal{G})-λ(G)){n\choose2}$ adjacencies. As special cases, this property describes all almost extremal graphs of order $n$ within $o(n^2)$ edges and shows that every extremal graph of order $n\ge n_0$ is a blow-up of $B$. We develop general methods for establishing stability-type results from flag algebra computations and apply them to concrete examples. In fact, one of our sufficient conditions for perfect stability is stated in a way that allows automatic verification by a computer. This gives a unifying way to obtain computer-assisted proofs of many new results.

math.CO↗
arXiv2017

Isometric copies of directed trees in orientations of graphs

For every $n\in\mathbb N$ we construct a finite graph $G$ such that every orientation $\vec G$ of $G$ contains an isometric copy of any oriented tree on $n$ vertices, and evaluate the smallest possible cardinality of $G$. On the other hand, we prove that every graph $G$ admits an orientation containing no directed $ω$-paths of infinite diameter.

math.CO↗
arXiv2016

The Erdős-Rothschild problem on edge-colourings with forbidden monochromatic cliques

Let $\mathbf{k} := (k_1,\dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges of colour $c$ contain no clique of order $k_c$. Write $F(n;\mathbf{k})$ to denote the maximum of $F(G;\mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by Erdős and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases. We prove that, for every $\mathbf{k}$ and $n$, there is a complete multipartite graph $G$ on $n$ vertices with $F(G;\mathbf{k}) = F(n;\mathbf{k})$. Also, for every $\mathbf{k}$ we construct a finite optimisation problem whose maximum is equal to the limit of $\log_2 F(n;\mathbf{k})/{n\choose 2}$ as $n$ tends to infinity. Our final result is a stability theorem for complete multipartite graphs $G$, describing the asymptotic structure of such $G$ with $F(G;\mathbf{k}) = F(n;\mathbf{k}) \cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem.

math.CO↗
arXiv2016

Supersaturation Problem for Color-Critical Graphs

The \emph{Turán function} $\ex(n,F)$ of a graph $F$ is the maximum number of edges in an $F$-free graph with $n$ vertices. The classical results of Turán and Rademacher from 1941 led to the study of supersaturated graphs where the key question is to determine $h_F(n,q)$, the minimum number of copies of $F$ that a graph with $n$ vertices and $\ex(n,F)+q$ edges can have. We determine $h_F(n,q)$ asymptotically when $F$ is \emph{color-critical} (that is, $F$ contains an edge whose deletion reduces its chromatic number) and $q=o(n^2)$. Determining the exact value of $h_F(n,q)$ seems rather difficult. For example, let $c_1$ be the limit superior of $q/n$ for which the extremal structures are obtained by adding some $q$ edges to a maximum $F$-free graph. The problem of determining $c_1$ for cliques was a well-known question of Erd\H os that was solved only decades later by Lovász and Simonovits. Here we prove that $c_1>0$ for every {color-critical}~$F$. Our approach also allows us to determine $c_1$ for a number of graphs, including odd cycles, cliques with one edge removed, and complete bipartite graphs plus an edge.

math.CO↗
arXiv2016

Measurable circle squaring

Laczkovich proved that if bounded subsets $A$ and $B$ of $R^k$ have the same non-zero Lebesgue measure and the box dimension of the boundary of each set is less than $k$, then there is a partition of $A$ into finitely many parts that can be translated to form a partition of $B$. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, this gives measurable and translation-only versions of Tarski's circle squaring and Hilbert's third problem.

math.MG↗
arXiv2016

Kőnig's Line Coloring and Vizing's Theorems for Graphings

The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge-coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by Kőnig, $d$ colors suffice if the graph is bipartite. We investigate the existence of measurable edge-colorings for graphings. A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence theory and measurable group theory. We show that every graphing of maximum degree $d$ admits a measurable edge-coloring with $d + O(\sqrt{d})$ colors; furthermore, if the graphing has no odd cycles, then $d+1$ colors suffice. In fact, if a certain conjecture about finite graphs that strengthens Vizing's theorem is true, then our method will show that $d+1$ colors are always enough.

math.CO↗
arXiv2015

The codegree threshold for 3-graphs with independent neighbourhoods

Given a family of 3-graphs $F$, we define its codegree threshold $\mathrm{coex}(n, F)$ to be the largest number $d=d(n)$ such that there exists an $n$-vertex 3-graph in which every pair of vertices is contained in at least $d$ 3-edges but which contains no member of $F$ as a subgraph. Let $F_{3,2}$ be the 3-graph on $\{a,b,c,d,e\}$ with 3-edges $\{abc,abd,abe,cde\}$. In this paper, we give two proofs that $\mathrm{coex}(n, F_{3,2})= n/3 +o(n)$, the first by a direct combinatorial argument and the second via a flag algebra computation. Information extracted from the latter proof is then used to obtain a stability result, from which in turn we derive the exact codegree threshold for all sufficiently large $n$: $\mathrm{coex}(n, F_{3,2})= \lfloor n/3 \rfloor -1$ if $n$ is congruent to $1$ modulo $3$, and $\lfloor n/3 \rfloor$ otherwise. In addition we determine the set of codegree-extremal configurations.

math.CO↗
arXiv2015

The maximal length of a gap between r-graph Turán densities

The Turán density $π(\cal F)$ of a family $\cal F$ of $r$-graphs is the limit as $n\to\infty$ of the maximum edge density of an $\cal F$-free $r$-graph on $n$ vertices. Erdos [Israel J. Math 2 (1964) 183--190] proved that no Turán density can lie in the open interval $(0,r!/r^r)$. Here we show that any other open subinterval of $[0,1]$ avoiding Turán densities has strictly smaller length. In particular, this implies a conjecture of Grosu [E-print arXiv:1403.4653v1, 2014].

math.CO↗
arXiv2015

Spherical sets avoiding a prescribed set of angles

Let $X$ be any subset of the interval $[-1,1]$. A subset $I$ of the unit sphere in $R^n$ will be called \emph{$X$-avoiding} if $ \notin X$ for any $u,v \in I$. The problem of determining the maximum surface measure of a $\{ 0 \}$-avoiding set was first stated in a 1974 note by Witsenhausen; there the upper bound of $1/n$ times the surface measure of the sphere is derived from a simple averaging argument. A consequence of the Frankl-Wilson theorem is that this fraction decreases exponentially, but until now the $1/3$ upper bound for the case $n=3$ has not moved. We improve this bound to $0.313$ using an approach inspired by Delsarte's linear programming bounds for codes, combined with some combinatorial reasoning. In the second part of the paper, we use harmonic analysis to show that for $n\geq 3$ there always exists an $X$-avoiding set of maximum measure. We also show with an example that a maximiser need not exist when $n=2$.

math.CO↗
arXiv2014

Monochromatic Clique Decompositions of Graphs

Let $G$ be a graph whose edges are coloured with $k$ colours, and $\mathcal H=(H_1,\dots , H_k)$ be a $k$-tuple of graphs. A monochromatic $\mathcal H$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms a monochromatic copy of $H_i$ in colour $i$, for some $1\le i\le k$. Let $ϕ_{k}(n,\mathcal H)$ be the smallest number $ϕ$, such that, for every order-$n$ graph and every $k$-edge-colouring, there is a monochromatic $\mathcal H$-decomposition with at most $ϕ$ elements. Extending the previous results of Liu and Sousa ["Monochromatic $K_r$-decompositions of graphs", Journal of Graph Theory}, 76:89--100, 2014], we solve this problem when each graph in $\mathcal H$ is a clique and $n\ge n_0(\mathcal H)$ is sufficiently large.

math.CO↗
arXiv2014

Coloring d-Embeddable k-Uniform Hypergraphs

This paper extends the scenario of the Four Color Theorem in the following way. Let H(d,k) be the set of all k-uniform hypergraphs that can be (linearly) embedded into R^d. We investigate lower and upper bounds on the maximum (weak and strong) chromatic number of hypergraphs in H(d,k). For example, we can prove that for d>2 there are hypergraphs in H(2d-3,d) on n vertices whose weak chromatic number is Omega(log n/log log n), whereas the weak chromatic number for n-vertex hypergraphs in H(d,d) is bounded by O(n^((d-2)/(d-1))) for d>2.

math.CO↗
arXiv2014

Minimum number of monotone subsequences of length 4 in permutations

We show that for every sufficiently large $n$, the number of monotone subsequences of length four in a permutation on $n$ points is at least $\binom{\lfloor n/3 \rfloor}{4} + \binom{\lfloor(n+1)/3\rfloor}{4} + \binom{\lfloor (n+2)/3\rfloor}{4}$. Furthermore, we characterize all permutations on $[n]$ that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colorings of complete graphs with two colors, where the number of monochromatic $K_4$'s is minimized. We show that all the extremal colorings must contain monochromatic $K_4$'s only in one of the two colors. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

math.CO↗