arXiv subjects

Alexey Pokrovskiy

Publications and source records attributed to Alexey Pokrovskiy.

At least 73 records · Page 4Linked to original sources

arXiv2016

Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph $K_n$ has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost $n$. In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured $K_n$ has a rainbow cycle of length $n-O(n^{3/4})$. One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured $K_n$ formed by the edges of a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular it has very good expansion properties.

math.CO↗

arXiv2016

Partitioning a graph into a cycle and a sparse graph

In this paper we investigate results of the form "every graph $G$ has a cycle $C$ such that the induced subgraph of $G$ on $V(G)\setminus V(C)$ has small maximum degree." Such results haven't been studied before, but are motivated by the Bessy and Thomassé Theorem which states that the vertices of any graph $G$ can be covered by a cycle $C_1$ in $G$ and disjoint cycle $C_2$ in the complement of $G$. There are two main theorems in this paper. The first is that every graph has a cycle with $Δ(G[V(G)\setminus V(C)])\leq \frac12(|V(G)\setminus V(C)|-1)$. The bound on the maximum degree $Δ(G[V(G)\setminus V(C)])$ is best possible. The second theorem is that every $k$-connected graph $G$ has a cycle with $Δ(G[V(G)\setminus V(C)])\leq \frac1{k+1}|V(G)\setminus V(C)|+3$. We also give an application of this second theorem to a conjecture about partitioning edge-coloured complete graphs into monochromatic cycles.

math.CO↗

arXiv2016

Ramsey goodness of paths

Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If graph $G$ is connected, it is well known and easy to show that $R(G,H) \geq (|G|-1)(χ(H)-1)+σ(H)$, where $χ(H)$ is the chromatic number of $H$ and $σ$ the size of the smallest color class in a $χ(H)$-coloring of $H$. A graph $G$ is called $H$-good if $R(G,H)= (|G|-1)(χ(H)-1)+σ(H)$. The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively studied since then. In this short note we prove that $n$-vertex path $P_n$ is $H$-good for all $n\geq 4|H|$. This proves in a strong form a conjecture of Allen, Brightwell, and Skokan.

math.CO↗

arXiv2016

Strong Ramsey Games: Drawing on an infinite board

We consider the strong Ramsey-type game $\mathcal{R}^{(k)}(\mathcal{H}, \aleph_0)$, played on the edge set of the infinite complete $k$-uniform hypergraph $K^k_{\mathbb{N}}$. Two players, called FP (the first player) and SP (the second player), take turns claiming edges of $K^k_{\mathbb{N}}$ with the goal of building a copy of some finite predetermined $k$-uniform hypergraph $\mathcal{H}$. The first player to build a copy of $\mathcal{H}$ wins. If no player has a strategy to ensure his win in finitely many moves, then the game is declared a draw. In this paper, we construct a $5$-uniform hypergraph $\mathcal{H}$ such that $\mathcal{R}^{(5)}(\mathcal{H}, \aleph_0)$ is a draw. This is in stark contrast to the corresponding finite game $\mathcal{R}^{(5)}(\mathcal{H}, n)$, played on the edge set of $K^5_n$. Indeed, using a classical game-theoretic argument known as \emph{strategy stealing} and a Ramsey-type argument, one can show that for every $k$-uniform hypergraph $\mathcal{G}$, there exists an integer $n_0$ such that FP has a winning strategy for $\mathcal{R}^{(k)}(\mathcal{G}, n)$ for every $n \geq n_0$.

math.CO↗

arXiv2015

On sets not belonging to algebras and rainbow matchings in graphs

Motivated by a question of Grinblat, we study the minimal number $\mathfrak{v}(n)$ that satisfies the following. If $A_1,\ldots, A_n$ are equivalence relations on a set $X$ such that for every $i\in[n]$ there are at least $\mathfrak{v}(n)$ elements whose equivalence classes with respect to $A_i$ are nontrivial, then $A_1, \ldots, A_n$ contain a rainbow matching, i.e. there exist $2n$ distinct elements $x_1,y_1,\ldots,x_n,y_n\in X$ with $x_i\sim_{A_i} y_i$ for each $i\in [n]$. Grinblat asked whether $\mathfrak{v}(n) = 3n-2$ for every $n\geq 4$. The best-known upper bound was $\mathfrak{v}(n) \leq 16n/5 + \mathcal{O}(1)$ due to Nivash and Omri. Transferring the problem into the setting of edge-coloured multigraphs, we affirm Grinblat's question asymptotically, i.e. we show that $\mathfrak{v}(n) = 3n+o(n)$.

math.CO↗

arXiv2015

Rainbow matchings and rainbow connectedness

Aharoni and Berger conjectured that every bipartite graph which is the union of n matchings of size n + 1 contains a rainbow matching of size n. This conjecture is a generalization of several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin squares. There have been many recent partial results about the Aharoni-Berger Conjecture. In the case when the matchings are much larger than n + 1, the best bound is currently due to Clemens and Ehrenmüller who proved the conjecture when the matchings are of size at least 3n/2 + o(n). When the matchings are all edge-disjoint and perfect, then the best result follows from a theorem of Häggkvist and Johansson which implies the conjecture when the matchings have size at least n + o(n). In this paper we show that the conjecture is true when the matchings have size n + o(n) and are all edge-disjoint (but not necessarily perfect). We also give an alternative argument to prove the conjecture when the matchings have size at least $ϕn + o(n)$ where $ϕ\approx 1.618$ is the Golden Ratio. Our proofs involve studying connectedness in coloured, directed graphs. The notion of connectedness that we introduce is new, and perhaps of independent interest.

math.CO↗

arXiv2014

Intersecting extremal constructions in Ryser's Conjecture for r-partite hypergraphs

Ryser's Conjecture states that for any $r$-partite $r$-uniform hypergraph the vertex cover number is at most $r-1$ times the matching number. This conjecture is only known to be true for $r\leq 3$. For intersecting hypergraphs, Ryser's Conjecture reduces to saying that the edges of every $r$-partite intersecting hypergraph can be covered by $r-1$ vertices. This special case of the conjecture has only been proven for $r \leq 5$. It is interesting to study hypergraphs which are extremal in Ryser's Conjecture i.e, those hypergraphs for which the vertex cover number is exactly $r-1$ times the matching number. There are very few known constructions of such graphs. For large $r$ the only known constructions come from projective planes and exist only when $r-1$ is a prime power. Mansour, Song and Yuster studied how few edges a hypergraph which is extremal for Ryser's Conjecture can have. They defined $f(r)$ as the minimum integer so that there exist an $r$-partite intersecting hypergraph $\mathcal{H}$ with $τ({\mathcal{H}}) = r -1$ and with $f(r)$ edges. They showed that $f(3) = 3, f(4) = 6$, $f(5) = 9$, and $12\leq f(6)\leq 15$. In this paper we focus on the cases when $r=6$ and 7. We show that $f(6)=13$ improving previous bounds. We also show that $f(7)\leq 22$, giving the first known extremal hypergraphs for the $r=7$ case of Ryser's Conjecture. These results have been obtained independently by Aharoni, Barat, and Wanless.

math.CO↗

arXiv2014

Graphs without proper subgraphs of minimum degree 3 and short cycles

We study graphs on $n$ vertices which have $2n-2$ edges and no proper induced subgraphs of minimum degree $3$. Erdős, Faudree, Gyárfás, and Schelp conjectured that such graphs always have cycles of lengths $3,4,5,\dots, C(n)$ for some function $C(n)$ tending to infinity. We disprove this conjecture, resolve a related problem about leaf-to-leaf path lengths in trees, and characterize graphs with $n$ vertices and $2n-2$ edges, containing no proper subgraph of minimum degree $3$.

math.CO↗

arXiv2014

Highly linked tournaments

A (possibly directed) graph is $k$-linked if for any two disjoint sets of vertices $\{x_1, \dots, x_k\}$ and $\{y_1, \dots, y_k\}$ there are vertex disjoint paths $P_1, \dots, P_k$ such that $P_i$ goes from $x_i$ to $y_{i}$. A theorem of Bollobás and Thomason says that every $22k$-connected (undirected) graph is $k$-linked. It is desirable to obtain analogues for directed graphs as well. Although Thomassen showed that the Bollobás-Thomason Theorem does not hold for general directed graphs, he proved an analogue of the theorem for tournaments - there is a function $f(k)$ such that every strongly $f(k)$-connected tournament is $k$-linked. The bound on $f(k)$ was reduced to $O(k \log k)$ by Kühn, Lapinskas, Osthus, and Patel, who also conjectured that a linear bound should hold. We prove this conjecture, by showing that every strongly $452k$-connected tournament is $k$-linked.

math.CO↗

arXiv2014

Edge disjoint Hamiltonian cycles in highly connected tournaments

Thomassen conjectured that there is a function $f(k)$ such that every strongly $f(k)$-connected tournament contains $k$ edge-disjoint Hamiltonian cycles. This conjecture was recently proved by Kühn, Lapinskas, Osthus, and Patel who showed that $f(k)\leq O(k^2(\log k)^2)$ and conjectured that there is a constant $C$ such that $f(k)\leq Ck^2$. We prove this conjecture.

math.CO↗

arXiv2014

Identifying codes and searching with balls in graphs

Given a graph $G$ and a positive integer $R$ we address the following combinatorial search theoretic problem: What is the minimum number of queries of the form "does an unknown vertex $v \in V(G)$ belong to the ball of radius $r$ around $u$?" with $u \in V(G)$ and $r\le R$ that is needed to determine $v$. We consider both the adaptive case when the $j$th query might depend on the answers to the previous queries and the non-adaptive case when all queries must be made at once. We obtain bounds on the minimum number of queries for hypercubes, the Erd\H os-Rényi random graphs and graphs of bounded maximum degree .

math.CO↗

arXiv2013

Calculating Ramsey numbers by partitioning coloured graphs

In this paper we prove a new result about partitioning coloured complete graphs and use it to determine certain Ramsey numbers exactly. The partitioning theorem we prove is that for k at least 1, in every edge colouring of a complete graph with the colours red and blue, it is possible to cover all the vertices with k disjoint red paths and a disjoint blue balanced complete (k+1)-partite graph. When the colouring is connected in red, we prove a stronger result - that it is possible to cover all the vertices with k red paths and a blue balanced complete (k+2)-partite graph. Using these results we determine the Ramsey number of a path on n vertices, versus a balanced complete k-partite graph, with m vertices in each part, whenever m-1 is divisible by n-1. This generalizes a result of Erdos who proved the m=1 case of this result. We also determine the Ramsey number of a path on n vertices versus the power of a path on n vertices. This solves a conjecture of Allen, Brightwell, and Skokan.

math.CO↗

arXiv2013

A linear bound on the Manickam-Miklos-Singhi Conjecture

Suppose that we have a set of numbers x_1, ..., x_n which have nonnegative sum. How many subsets of k numbers from {x_1, ..., x_n} must have nonnegative sum? Manickam, Miklos, and Singhi conjectured that for n at least 4k the answer is (n-1 \choose k-1). This conjecture is known to hold when n is large compared to k. The best known bounds are due to Alon, Huang, and Sudakov who proved the conjecture when n > 33k^2. In this paper we improve this bound by showing that there is a constant C such that the conjecture holds when n > Ck.

math.CO↗

arXiv2013

Advantage in the discrete Voronoi game

We study the discrete Voronoi game, where two players alternately claim vertices of a graph for t rounds. In the end, the remaining vertices are divided such that each player receives the vertices that are closer to his or her claimed vertices. We prove that there are graphs for which the second player gets almost all vertices in this game, but this is not possible for bounded-degree graphs. For trees, the first player can get at least one quarter of the vertices, and we give examples where she can get only little more than one third of them. We make some general observations, relating the result with many rounds to the result for the one-round game on the same graph.

math.CO↗

arXiv2012

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

A conjecture of Erdős, Gyárfás, and Pyber says that in any edge-colouring of a complete graph with r colours, it is possible to cover all the vertices with r vertex-disjoint monochromatic cycles. So far, this conjecture has been proven only for r = 2. In this paper we show that in fact this conjecture is false for all r > 2. In contrast to this, we show that in any edge-colouring of a complete graph with three colours, it is possible to cover all the vertices with three vertex-disjoint monochromatic paths, proving a particular case of a conjecture due to Gyárfás. As an intermediate result we show that in any edge-colouring of the complete graph with the colours red and blue, it is possible to cover all the vertices with a red path, and a disjoint blue balanced complete bipartite graph.

math.CO↗

arXiv2012

Edge growth in graph powers

For a graph G, its rth power G^r has the same vertex set as G, and has an edge between any two vertices within distance r of each other in G. We give a lower bound for the number of edges in the rth power of G in terms of the order of G and the minimal degree of G. As a corollary we determine how small the ratio e(G^r)/e(G) can be for regular graphs of diameter at least r.

math.CO↗

arXiv2011

Periodic Sequences of Arbitrage: A Tale of Four Currencies

This paper investigates arbitrage chains involving four currencies and four foreign exchange trader-arbitrageurs. In contrast with the three-currency case, we find that arbitrage operations when four currencies are present may appear periodic in nature, and not involve smooth convergence to a "balanced" ensemble of exchange rates in which the law of one price holds. The goal of this article is to understand some interesting features of sequences of arbitrage operations, features which might well be relevant in other contexts in finance and economics.

q-fin.GN↗

arXiv2010

Growth of graph powers

For a graph G, its rth power is constructed by placing an edge between two vertices if they are within distance r of each other. In this note we study the amount of edges added to a graph by taking its rth power. In particular we obtain that either the rth power is complete or "many" new edges are added. This is an extension of a result obtained by P. Hegarty for cubes of graphs.

math.CO↗