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Publisher: Cambridge University Press   (Total: 371 journals)

 Combinatorics, Probability and ComputingJournal Prestige (SJR): 0.839 Citation Impact (citeScore): 1Number of Followers: 4      Hybrid journal (It can contain Open Access articles) ISSN (Print) 0963-5483 - ISSN (Online) 1469-2163 Published by Cambridge University Press  [371 journals]
• Introduction
• Authors: PAUL BALISTER; BÉLA BOLLOBÁS, IMRE LEADER, ROB MORRIS, OLIVER RIORDAN
Pages: 441 - 441
Abstract: This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from the 17th to the 23rd April 2016. The lectures at this meeting focused on the common themes of Combinatorics and Discrete Probability, with many of the problems studied originating in Theoretical Computer Science. The lectures, many of which were given by young participants, stimulated fruitful discussions. The fact that the participants work in different and yet related topics, and the open problems session held during the meeting, encouraged interesting discussions and collaborations.
PubDate: 2018-07-01T00:00:00.000Z
DOI: 10.1017/S0963548318000317
Issue No: Vol. 27, No. 4 (2018)

• Uniformly Discrete Forests with Poor Visibility
• Authors: NOGA ALON
Pages: 442 - 448
Abstract: We prove that there is a set F in the plane so that the distance between any two points of F is at least 1, and for any positive ϵ < 1, and every line segment in the plane of length at least ϵ−1−o(1), there is a point of F within distance ϵ of the segment. This is tight up to the o(1)-term in the exponent, improving earlier estimates of Peres, of Solomon and Weiss, and of Adiceam.
PubDate: 2018-07-01T00:00:00.000Z
DOI: 10.1017/S0963548317000505
Issue No: Vol. 27, No. 4 (2018)

• Triangle-Tilings in Graphs Without Large Independent Sets
• Authors: JÓZSEF BALOGH; ANDREW McDOWELL, THEODORE MOLLA, RICHARD MYCROFT
Pages: 449 - 474
Abstract: We study the minimum degree necessary to guarantee the existence of perfect and almost-perfect triangle-tilings in an n-vertex graph G with sublinear independence number. In this setting, we show that if δ(G) ≥ n/3 + o(n), then G has a triangle-tiling covering all but at most four vertices. Also, for every r ≥ 5, we asymptotically determine the minimum degree threshold for a perfect triangle-tiling under the additional assumptions that G is Kr-free and n is divisible by 3.
PubDate: 2018-07-01T00:00:00.000Z
DOI: 10.1017/S0963548318000196
Issue No: Vol. 27, No. 4 (2018)

• Packing Hamilton Cycles Online
• Authors: JOSEPH BRIGGS; ALAN FRIEZE, MICHAEL KRIVELEVICH, PO-SHEN LOH, BENNY SUDAKOV
Pages: 475 - 495
Abstract: It is known that w.h.p. the hitting time τ2σ for the random graph process to have minimum degree 2σ coincides with the hitting time for σ edge-disjoint Hamilton cycles [4, 9, 13]. In this paper we prove an online version of this property. We show that, for a fixed integer σ ⩾ 2, if random edges of Kn are presented one by one then w.h.p. it is possible to colour the edges online with σ colours so that at time τ2σ each colour class is Hamiltonian.
PubDate: 2018-07-01T00:00:00.000Z
DOI: 10.1017/S0963548318000159
Issue No: Vol. 27, No. 4 (2018)

• k-SAT+Formulas&rft.title=Combinatorics,+Probability+and+Computing&rft.issn=0963-5483&rft.date=2018&rft.volume=27&rft.spage=496&rft.epage=530&rft.aulast=COJA-OGHLAN&rft.aufirst=AMIN&rft.au=AMIN+COJA-OGHLAN&rft.au=NICK+WORMALD&rft_id=info:doi/10.1017/S0963548318000263">The Number of Satisfying Assignments of Random Regular        class="italic">k-SAT Formulas
• Authors: AMIN COJA-OGHLAN; NICK WORMALD
Pages: 496 - 530
Abstract: Let Φ be a random k-SAT formula in which every variable occurs precisely d times positively and d times negatively. Assuming that k is sufficiently large and that d is slightly below the critical degree where the formula becomes unsatisfiable with high probability, we determine the limiting distribution of the number of satisfying assignments.
PubDate: 2018-07-01T00:00:00.000Z
DOI: 10.1017/S0963548318000263
Issue No: Vol. 27, No. 4 (2018)

• The Minimum Number of Edges in Uniform Hypergraphs with Property O
• Authors: DWIGHT DUFFUS; BILL KAY, VOJTĚCH RÖDL
Pages: 531 - 538
Abstract: An oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if, for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the minimum number of edges in a hypergraph with Property O.
PubDate: 2018-07-01T00:00:00.000Z
DOI: 10.1017/S096354831700058X
Issue No: Vol. 27, No. 4 (2018)

• Fast Property Testing and Metrics for Permutations
• Authors: JACOB FOX; FAN WEI
Pages: 539 - 579
Abstract: The goal of property testing is to quickly distinguish between objects which satisfy a property and objects that are ε-far from satisfying the property. There are now several general results in this area which show that natural properties of combinatorial objects can be tested with ‘constant’ query complexity, depending only on ε and the property, and not on the size of the object being tested. The upper bound on the query complexity coming from the proof techniques is often enormous and impractical. It remains a major open problem if better bounds hold.Maybe surprisingly, for testing with respect to the rectangular distance, we prove there is a universal (not depending on the property), polynomial in 1/ε query complexity bound for two-sided testing hereditary properties of sufficiently large permutations. We further give a nearly linear bound with respect to a closely related metric which also depends on the smallest forbidden subpermutation for the property. Finally, we show that several different permutation metrics of interest are related to the rectangular distance, yielding similar results for testing with respect to these metrics.
PubDate: 2018-07-01T00:00:00.000Z
DOI: 10.1017/S096354831800024X
Issue No: Vol. 27, No. 4 (2018)

• Minimizing the Number of Triangular Edges
• Authors: VYTAUTAS GRUSLYS; SHOHAM LETZTER
Pages: 580 - 622
Abstract: We consider the problem of minimizing the number of edges that are contained in triangles, among n-vertex graphs with a given number of edges. For sufficiently large n, we prove an exact formula for this minimum, which partially resolves a conjecture of Füredi and Maleki.
PubDate: 2018-07-01T00:00:00.000Z
DOI: 10.1017/S0963548317000189
Issue No: Vol. 27, No. 4 (2018)

• Anagram-Free Colourings of Graphs
• Authors: NINA KAMČEV; TOMASZ ŁUCZAK, BENNY SUDAKOV
Pages: 623 - 642
Abstract: A sequence S is called anagram-free if it contains no consecutive symbols r1r2. . .rkrk+1. . .r2k such that rk+1. . .r2k is a permutation of the block r1r2. . .rk. Answering a question of Erdős and Brown, Keränen constructed an infinite anagram-free sequence on four symbols. Motivated by the work of Alon, Grytczuk, Hałuszczak and Riordan [2], we consider a natural generalization of anagram-free sequences for graph colourings. A colouring of the vertices of a given graph G is called anagram-free if the sequence of colours on any path in G is anagram-free. We call the minimal number of colours needed for such a colouring the anagram-chromatic number of G.In this paper we study the anagram-chromatic number of several classes of graphs like trees, minor-free graphs and bounded-degree graphs. Surprisingly, we show that there are bounded-degree graphs (such as random regular graphs) in which anagrams cannot be avoided unless we essentially give each vertex a separate colour.
PubDate: 2018-07-01T00:00:00.000Z
DOI: 10.1017/S096354831700027X
Issue No: Vol. 27, No. 4 (2018)

• Drift Analysis and Evolutionary Algorithms Revisited
• Authors: J. LENGLER; A. STEGER
Pages: 643 - 666
Abstract: One of the easiest randomized greedy optimization algorithms is the following evolutionary algorithm which aims at maximizing a function f: {0,1}n → ℝ. The algorithm starts with a random search point ξ ∈ {0,1}n, and in each round it flips each bit of ξ with probability c/n independently at random, where c > 0 is a fixed constant. The thus created offspring ξ' replaces ξ if and only if f(ξ') ≥ f(ξ). The analysis of the runtime of this simple algorithm for monotone and for linear functions turned out to be highly non-trivial. In this paper we review known results and provide new and self-contained proofs of partly stronger results.
PubDate: 2018-07-01T00:00:00.000Z
DOI: 10.1017/S0963548318000275
Issue No: Vol. 27, No. 4 (2018)

• Connections in Randomly Oriented Graphs
• Authors: BHARGAV NARAYANAN
Pages: 667 - 671
Abstract: Given an undirected graph G, let us randomly orient G by tossing independent (possibly biased) coins, one for each edge of G. Writing a → b for the event that there exists a directed path from a vertex a to a vertex b in such a random orientation, we prove that for any three vertices s, a and b of G, we have ℙ(s → a ∩ s → b) ⩾ ℙ(s → a) ℙ(s → b).
PubDate: 2018-07-01T00:00:00.000Z
DOI: 10.1017/S0963548316000341
Issue No: Vol. 27, No. 4 (2018)

• The Probability of Non-Existence of a Subgraph in a Moderately Sparse
Random Graph
• Authors: DUDLEY STARK; NICK WORMALD
Pages: 672 - 715
Abstract: We develop a general procedure that finds recursions for statistics counting isomorphic copies of a graph G0 in the common random graph models ${\cal G}$ (n,m) and ${\cal G}$ (n,p). Our results apply when the average degrees of the random graphs are below the threshold at which each edge is included in a copy of G0. This extends an argument given earlier by the second author for G0=K3 with a more restricted range of average degree. For all strictly balanced subgraphs G0, our results give much information on the distribution of the number of copies of G0 that are not in large ‘clusters’ of copies. The probability that a random graph in ${\cal G}$ (n,p) has no copies of G0 is shown to be given asymptotically by the exponential of a power series in n and p, over a fairly wide range of p. A corresponding result is also given for ${\cal G}$ (n,m), which gives an asymptotic formula for the number of graphs with n vertices, m edges and no copies of G0, for the applicable range of m. An example is given, computing the asymptotic probability that a random graph has no triangles for p=o(n−7/11) in ${\cal G}$ (n,p) and for m=o(n15/11) in ${\cal G}$ (n,m), extending results of the second author.
PubDate: 2018-07-01T00:00:00.000Z
DOI: 10.1017/S0963548318000202
Issue No: Vol. 27, No. 4 (2018)

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