arXiv subjects

Klas Markström

Publications and source records attributed to Klas Markström.

At least 37 records · Page 2Linked to original sources

arXiv2021

Avoiding and extending partial edge colorings of hypercubes

We consider the problem of extending and avoiding partial edge colorings of hypercubes; that is, given a partial edge coloring $φ$ of the $d$-dimensional hypercube $Q_d$, we are interested in whether there is a proper $d$-edge coloring of $Q_d$ that agrees with the coloring $φ$ on every edge that is colored under $φ$; or, similarly, if there is a proper $d$-edge coloring that disagrees with $φ$ on every edge that is colored under $φ$. In particular, we prove that for any $d\geq 1$, if $φ$ is a partial $d$-edge coloring of $Q_d$, then $φ$ is avoidable if every color appears on at most $d/8$ edges and the coloring satisfies a relatively mild structural condition, or $φ$ is proper and every color appears on at most $d-2$ edges. We also show that the same conclusion holds if $d$ is divisible by $3$ and every color class of $φ$ is an induced matching. Moreover, for all $1 \leq k \leq d$, we characterize for which configurations consisting of a partial coloring $φ$ of $d-k$ edges and a partial coloring $ψ$ of $k$ edges, there is an extension of $φ$ that avoids $ψ$.

math.CO↗

arXiv2020

Existence thresholds and Ramsey properties of random posets

Let $\mathcal P(n)$ denote the power set of $[n]$, ordered by inclusion, and let $\mathcal P (n,p)$ denote the random poset obtained from $\mathcal P(n)$ by retaining each element from $\mathcal P (n)$ independently at random with probability $p$ and discarding it otherwise. Given any fixed poset $F$ we determine the threshold for the property that $\mathcal P(n,p)$ contains $F$ as an induced subposet. We also asymptotically determine the number of copies of a fixed poset $F$ in $\mathcal P(n)$. Finally, we obtain a number of results on the Ramsey properties of the random poset $\mathcal P(n,p)$.

math.CO↗

arXiv2020

Partite Turán-densities for complete $r-$uniform hypergraphs on $r+1$ vertices

In this paper we investigate density conditions for finding a complete $r$-uniform hypergraph $K_{r+1}^{(r)}$ on $r+1$ vertices in an $(r+1)$-partite $r$-uniform hypergraph $G$. First we prove an optimal condition in terms of the densities of the $(r+1)$ induced $r$-partite subgraphs of $G$. Second, we prove a version of this result where we assume that $r$-tuples of vertices in $G$ have their neighbours evenly distributed in $G$. Third, we also prove a counting result for the minimum number of copies of $K_{r+1}^{(r)}$ when $G$ satisfies our density bound, and present some open problems. A striking difference between the graph, $r=2$, and the hypergraph, $ r \geq 3 $, cases is that in the first case both the existence threshold and the counting function are non-linear in the involved densities, whereas for hypergraphs they are given by a linear function. Also, the smallest density of the $r$-partite parts needed to ensure the existence of a complete $r$-graph with $(r+1)$ vertices is equal to the golden ratio $τ=0.618\ldots$ for $r=2$, while it is $\frac{r}{r+1}$for $r\geq3$.

math.CO↗

arXiv2020

Discriminating Non-Isomorphic Graphs with an Experimental Quantum Annealer

We demonstrate experimentally the ability of a quantum annealer to distinguish between sets of non-isomorphic graphs that share the same classical Ising spectrum. Utilizing the pause-and-quench features recently introduced into D-Wave quantum annealing processors, which allow the user to probe the quantum Hamiltonian realized in the middle of an anneal, we show that obtaining thermal averages of diagonal observables of classically indistinguishable non-isomorphic graphs encoded into transverse-field Ising Hamiltonians enable their discrimination. We discuss the significance of our results in the context of the graph isomorphism problem.

quant-ph↗

arXiv2020

Random Uniform and Pure Random Simplicial Complexes

In this paper we introduce a method which allows us to study properties of the random uniform simplicial complex. That is, we assign equal probability to all simplicial complexes with a given number of vertices and then consider properties of a complex under this measure. We are able to determine or present bounds for a number of topological and combinatorial properties. We also study the random pure simplicial complex of dimension $d$, generated by letting any subset of size $d+1$ of a set of $n$ vertices be a facet with probability $p$ and considering the simplicial complex generated by these facets. We compare the behaviour of these models for suitable values of $d$ and $p$. Finally we use the equivalence between simplicial complexes and monotone boolean functions to study the behaviour of typical such functions. Specifically we prove that most monotone boolean functions are evasive, hence proving that the well known Evasiveness conjecture is generically true for monotone boolean functions without symmetry assumptions.

math.CO↗

arXiv2019

Biased random k-SAT

The basic random $k$-SAT problem is: Given a set of $n$ Boolean variables, and $m$ clauses of size $k$ picked uniformly at random from the set of all such clauses on our variables, is the conjunction of these clauses satisfiable? Here we consider a variation of this problem where there is a bias towards variables occurring positive -- i.e. variables occur negated w.p. $0<p< \frac{1}{2}$ and positive otherwise -- and study how the satisfiability threshold depends on $p$. For $p<\frac{1}{2}$ this model breaks many of the symmetries of the original random $k$-SAT problem, e.g. the distribution of satisfying assignments in the Boolean cube is no longer uniform. For any fixed $k$, we find the asymptotics of the threshold as $p$ approaches $0$ or $\frac{1}{2}$. The former confirms earlier predictions based on numerical studies and heuristic methods from statistical physics.

math.CO↗

arXiv2019

Triangle-degrees in graphs and tetrahedron coverings in 3-graphs

We investigate a covering problem in $3$-uniform hypergraphs ($3$-graphs): given a $3$-graph $F$, what is $c_1(n,F)$, the least integer $d$ such that if $G$ is an $n$-vertex $3$-graph with minimum vertex degree $δ_1(G)>d$ then every vertex of $G$ is contained in a copy of $F$ in $G$ ? We asymptotically determine $c_1(n,F)$ when $F$ is the generalised triangle $K_4^{(3)-}$, and we give close to optimal bounds in the case where $F$ is the tetrahedron $K_4^{(3)}$ (the complete $3$-graph on $4$ vertices). This latter problem turns out to be a special instance of the following problem for graphs: given an $n$-vertex graph $G$ with $m> n^2/4$ edges, what is the largest $t$ such that some vertex in $G$ must be contained in $t$ triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.

math.CO↗

arXiv2018

Triples of Orthogonal Latin and Youden Rectangles For Small Orders

We have performed a complete enumeration of non-isotopic triples of mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$. Here we will present a census of such triples, classified by various properties, including the order of the autotopism group of the triple. As part of this we have also achieved the first enumeration of pairwise orthogonal triples of Youden rectangles. We have also studied orthogonal triples of $k \times 8$ rectangles which are formed by extending mutually orthogonal triples with non-trivial autotopisms one row at a time, and requiring that the autotopism group is non-trivial in each step. This class includes a triple coming from the projective plane of order 8. Here we find a remarkably symmetrical pair of triples of $4 \times 8$ rectangles, formed by juxtaposing two selected copies of complete sets of MOLS of order 4.

math.CO↗

arXiv2018

Latin Cubes with Forbidden Entries

We consider the problem of constructing Latin cubes subject to the condition that some symbols may not appear in certain cells. We prove that there is a constant $γ> 0$ such that if $n=2^k$ and $A$ is $3$-dimensional $n\times n\times n$ array where every cell contains at most $γn$ symbols, and every symbol occurs at most $γn$ times in every line of $A$, then $A$ is {\em avoidable}; that is, there is a Latin cube $L$ of order $n$ such that for every $1\leq i,j,k\leq n$, the symbol in position $(i,j,k)$ of $L$ does not appear in the corresponding cell of $A$.

math.CO↗

arXiv2016

Full subgraphs

Let $G=(V,E)$ be a graph of density $p$ on $n$ vertices. Following Erdős, Łuczak and Spencer, an $m$-vertex subgraph $H$ of $G$ is called {\em full} if $H$ has minimum degree at least $p(m - 1)$. Let $f(G)$ denote the order of a largest full subgraph of $G$. If $p\binom{n}{2}$ is a non-negative integer, define \[ f(n,p) = \min\{f(G) : \vert V(G)\vert = n, \ \vert E(G)\vert = p\binom{n}{2} \}.\] Erdős, Łuczak and Spencer proved that for $n \geq 2$, \[ (2n)^{\frac{1}{2}} - 2 \leq f(n, {\frac{1}{2}}) \leq 4n^{\frac{2}{3}}(\log n)^{\frac{1}{3}}.\] In this paper, we prove the following lower bound: for $n^{-\frac{2}{3}} <p_n <1-n^{-\frac{1}{7}}$, \[ f(n,p) \geq \frac{1}{4}(1-p)^{\frac{2}{3}}n^{\frac{2}{3}} -1.\] Furthermore we show that this is tight up to a multiplicative constant factor for infinitely many $p$ near the elements of $\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\dots\}$. In contrast, we show that for any $n$-vertex graph $G$, either $G$ or $G^c$ contains a full subgraph on $Ω(\frac{n}{\log n})$ vertices. Finally, we discuss full subgraphs of random and pseudo-random graphs, and several open problems.

math.CO↗

arXiv2016

Restricted completion of sparse partial Latin squares

An $n \times n$ partial Latin square $P$ is called $α$-dense if each row and column has at most $αn$ non-empty cells and each symbol occurs at most $αn$ times in $P$. An $n \times n$ array $A$ where each cell contains a subset of $\{1,\dots, n\}$ is a $(βn, βn, βn)$-array if each symbol occurs at most $βn$ times in each row and column and each cell contains a set of size at most $βn$. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants $α, β> 0$ such that, for every positive integer $n$, if $P$ is an $α$-dense $n \times n$ partial Latin square, $A$ is an $n \times n$ $(βn, βn, βn)$-array, and no cell of $P$ contains a symbol that appears in the corresponding cell of $A$, then there is a completion of $P$ that avoids $A$; that is, there is a Latin square $L$ that agrees with $P$ on every non-empty cell of $P$, and, for each $i,j$ satisfying $1 \leq i,j \leq n$, the symbol in position $(i,j)$ in $L$ does not appear in the corresponding cell of $A$.

math.CO↗

arXiv2016

Speed and concentration of the covering time for structured coupon collectors

Let $V$ be an $n$-set, and let $X$ be a random variable taking values in the powerset of $V$. Suppose we are given a sequence of random coupons $X_1, X_2, \ldots $, where the $X_i$ are independent random variables with distribution given by $X$. The covering time $T$ is the smallest integer $t\geq 0$ such that $\bigcup_{i=1}^tX_i=V$. The distribution of $T$ is important in many applications in combinatorial probability, and has been extensively studied. However the literature has focussed almost exclusively on the case where $X$ is assumed to be symmetric and/or uniform in some way. In this paper we study the covering time for much more general random variables $X$; we give general criteria for $T$ being sharply concentrated around its mean, precise tools to estimate that mean, as well as examples where $T$ fails to be concentrated and when structural properties in the distribution of $X$ allow for a very different behaviour of $T$ relative to the symmetric/uniform case.

math.PR↗

arXiv2015

From the Ising and Potts models to the general graph homomorphism polynomial

In this note we study some of the properties of the generating polynomial for homomorphisms from a graph to at complete weighted graph on $q$ vertices. We discuss how this polynomial relates to a long list of other well known graph polynomials and the partition functions for different spin models, many of which are specialisations of the homomorphism polynomial. We also identify the smallest graphs which are not determined by their homomorphism polynomials for $q=2$ and $q=3$ and compare this with the corresponding minimal examples for the $U$-polynomial, which generalizes the well known Tutte-polynomal.

math.CO↗

arXiv2015

Problem collection from the IML programme: Graphs, Hypergraphs, and Computing

This collection of problems and conjectures is based on a subset of the open problems from the seminar series and the problem sessions of the Institut Mitag-Leffler programme Graphs, Hypergraphs, and Computing. Each problem contributor has provided a write up of their proposed problem and the collection has been edited by Klas Markström.

math.CO↗

arXiv2015

Random subcube intersection graphs I: cliques and covering

We study random subcube intersection graphs, that is, graphs obtained by selecting a random collection of subcubes of a fixed hypercube $Q_d$ to serve as the vertices of the graph, and setting an edge between a pair of subcubes if their intersection is non-empty. Our motivation for considering such graphs is to model `random compatibility' between vertices in a large network. For both of the models considered in this paper, we determine the thresholds for covering the underlying hypercube $Q_d$ and for the appearance of s-cliques. In addition we pose some open problems.

math.PR↗

arXiv2014

The straight line complexity of small factorials and primorials

In this paper we determine the straight-line complexity of $n!$ for $n\leq 22$ and give bounds for the complexities up to $n=46$. In the same way we determine the straight-line complexity of the product of the first primes up to $p=31$ and gives bounds for $p\leq 43$. Our results are based on an exhaustive computer search of the short length straight-line programs.

cs.CC↗

arXiv2014

Perfect matchings in 3-partite 3-uniform hypergraphs

Let $H$ be a $3$-partite $3$-uniform hypergraph, i.e. a $3$-uniform hypergraph such that every edge intersects every partition class in exactly one vertex, with each partition class of size $n$. We determine a Dirac-type vertex degree threshold for perfect matchings in $3$-partite $3$-uniform hypergraphs.

math.CO↗

arXiv2014

$\ell$-degree Turán density

Let $H_n$ be a $k$-graph on $n$ vertices. For $0 \le \ell \ell >1$, the set of $π_{\ell}^k(\mathcal{F})$ is dense in the interval $[0,1)$. Hence, there is no "jump" for $\ell$-degree Turán density when $k>\ell >1$. We also give a lower bound on $π_{\ell}^k(\mathcal{F})$ in terms of an ordinary Turán density.

math.CO↗