The approximate Loebl-Koml\'os-S\'os Conjecture IV: Embedding techniques and the proof of the main result

This is the last paper of a series of four papers in which we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number~$k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first two papers of this series, we decomposed the host graph $G$, and found a suitable combinatorial structure inside the decomposition. In the third paper, we refined this structure, and proved that any graph satisfying the conditions of the above approximate version of the Loebl-Komlos-Sos Conjecture contains one of ten specific configurations. In this paper we embed the tree $T$ in each of the ten configurations.Comment: 81 pages, 12 figures. A fix reflecting the change of Preconfiguration Clubs in Paper III, additional small change

The Approximate Loebl-Koml\'os-S\'os Conjecture III: The finer structure of LKS graphs

This is the third of a series of four papers in which we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first paper of the series, we gave a decomposition of the graph $G$ into several parts of different characteristics. In the second paper, we found a combinatorial structure inside the decomposition. In this paper, we will give a refinement of this structure. In the forthcoming fourth paper, the refined structure will be used for embedding the tree $T$.Comment: 59 pages, 4 figures; further comments by a referee incorporated; this includes a subtle but important fix to Lemma 5.1; as a consequence, Preconfiguration Clubs was change

The approximate Loebl-Koml\'os-S\'os Conjecture II: The rough structure of LKS graphs

This is the second of a series of four papers in which we prove the following relaxation of the Loebl-Komlos--Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first paper of the series, we gave a decomposition of the graph $G$ into several parts of different characteristics; this decomposition might be viewed as an analogue of a regular partition for sparse graphs. In the present paper, we find a combinatorial structure inside this decomposition. In the last two papers, we refine the structure and use it for embedding the tree $T$.Comment: 38 pages, 4 figures; new is Section 5.1.1; accepted to SIDM

Z historickej slovenskej hydronymie a ojkonymie – Nitrava èi Nitra?

V príspevku autor analyzuje historické slovenské toponymum Nitra / Nitrava, ktorý podnes v slovenskom jazyku poznáme ako hydonymum a ojkonymum. Autor prinása ostatné výklady tohto slova od rôznych jazykovedcov onomastikov, ktorí vysvetlovali jeho pôvod z predslovanského indoeurópskeho základu s významom "voda". Starsie výklady nachádzame v klasickom a monumentálnom diele V. Smilauera. V prvom prípade môzeme uvazovato poèiatoènom pouzití slova ako hydronyma, v druhom prípade o jeho pôvodnom pouzití ako ojkonyma (pomenovanie obyvatelstva na vypálenom mieste). Historický formant -ava v hydronymii sa tradiène vysvetluje zo starogerm. slova ahwa, no je frekventovaný v lexike vsetkých slovanských jazykov. V súèasnej slovenskej hydronymii ho nachádzme v názvoch riek Orava, Ondava a pod. V stúdii analyzujeme aj moznosti vplyvu maïarského jazyka na zmenu formantu -ava > -va v 10. - 12. storoèí a ukazuje sa, ze pôvodná forma hydronyma bola Nitra.В статье анализируется древний словацкий топоним Нитра/Нитрава, который до сих пор выступает в словацком языке в качестве гидронима и ойконима. Автор приводит толкование этого слова разными лингвистами . ономастами, которые объясняли его происхождение от дославянской основы со значением “вода”. В первом случае внимание обращено на первичное использование слова как гидронима. Во втором случае на его первичное использование как ойконима. Исторический формант -ава в гидронимии традиционно толкуется из др.герм. ahwa, но он широко распространен в лексике всех славянских языков. В Словакии находим его в названиях Орава, Ондава и тд. Анализируется возможности влияния венгерского языка на изменение форманта -ава > -ва в 10 - 12 вв.In the paper the author analyzes Slovak historical toponym Nitra / Nitrava which up to the present has been known in the Slovak language as a hydronym and ojconym. The author presents the latest explanations of this word from different linguists onomastics who explained its origin from the pre-Slavic Indoeuropean basis with the meaning «water». We find the older explanations in the classic and monumental work of V. Smilauer. In the first case we can think about the initial use of the word as a hydronym, in the second case about its original use as an ojconym /naming the population in the burnt place/. Historical formant ava is in hydronymy traditionally explained from Old German ahwa but it is frequent in lexicon of all Slavic languages. We find it in the contemporary Slovak hydronymy in the names of the rivers Orava, Ondava, etc. In the essay we analyze also the possibilities of the Hungarian language influence upon the change of the formant -ava > -va in 10 - 12 century and it is obvious that the original form of the hydronym was Nitra

Stable divisorial gonality is in NP

Divisorial gonality and stable divisorial gonality are graph parameters, which have an origin in algebraic geometry. Divisorial gonality of a connected graph $G$ can be defined with help of a chip firing game on $G$. The stable divisorial gonality of $G$ is the minimum divisorial gonality over all subdivisions of edges of $G$. In this paper we prove that deciding whether a given connected graph has stable divisorial gonality at most a given integer $k$ belongs to the class NP. Combined with the result that (stable) divisorial gonality is NP-hard by Gijswijt, we obtain that stable divisorial gonality is NP-complete. The proof consist of a partial certificate that can be verified by solving an Integer Linear Programming instance. As a corollary, we have that the number of subdivisions needed for minimum stable divisorial gonality of a graph with $n$ vertices is bounded by $2^{p(n)}$ for a polynomial $p$

The approximate Loebl-Komlós-Sós conjecture I: The sparse decomposition

In a series of four papers we prove the following relaxation of the Loebl-Komlós-Sós conjecture: For every α > 0 there exists a number k0 such that for every k > k0, every n-vertex graph G with at least (1/2 + α)n vertices of degree at least (1 + α)k contains each tree T of order k as a subgraph. The method to prove our result follows a strategy similar to approaches that employ the Szemerédi regularity lemma: We decompose the graph G, find a suitable combinatorial structure inside the decomposition, and then embed the tree T into G using this structure. Since for sparse graphs G, the decomposition given by the regularity lemma is not helpful, we use a more general decomposition technique. We show that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. In this paper, we introduce this novel decomposition technique. In the three follow-up papers, we find a suitable combinatorial structure inside the decomposition, which we then use for embedding the tree. © 2017 the authors

Quantum-state synthesis of multi-mode bosonic fields: Preparation of arbitrary states of 2-D vibrational motion of trapped ions

We present a universal algorithm for an efficient deterministic preparation of an arbitrary two--mode bosonic state. In particular, we discuss in detail preparation of entangled states of a two-dimensional vibrational motion of a trapped ion via a sequence of laser stimulated Raman transitions. Our formalism can be generalized for multi-mode bosonic fields. We examine stability of our algorithm with respect to a technical noise.Comment: 8 pages, revtex, including 2 ps-figures, section about physical implementation added, references updated, submitted to Phys. Rev. A, computer program available at http://www.savba.sk/sav/inst/fyzi/qo

Quantum synthesis of arbitrary unitary operators

Nature provides us with a restricted set of microscopic interactions. The question is whether we can synthesize out of these fundamental interactions an arbitrary unitary operator. In this paper we present a constructive algorithm for realization of any unitary operator which acts on a (truncated) Hilbert space of a single bosonic mode. In particular, we consider a physical implementation of unitary transformations acting on 1-dimensional vibrational states of a trapped ion. As an example we present an algorithm which realizes the discrete Fourier transform.Comment: 6 RevTeX pages with 3 figures, submitted to Phys.Rev.A, see also http://nic.savba.sk/sav/inst/fyzi/qo