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  • A quantitative Lovász criterion for Property B
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-08-07
    Asaf Ferber; Asaf Shapira

    A well-known observation of Lovász is that if a hypergraph is not 2-colourable, then at least one pair of its edges intersect at a single vertex. In this short paper we consider the quantitative version of Lovász’s criterion. That is, we ask how many pairs of edges intersecting at a single vertex should belong to a non-2-colourable n-uniform hypergraph. Our main result is an exact answer to this question

    更新日期:2020-08-08
  • Deterministic counting of graph colourings using sequences of subgraphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-06-22
    Charilaos Efthymiou

    In this paper we propose a polynomial-time deterministic algorithm for approximately counting the k-colourings of the random graph G(n, d/n), for constant d>0. In particular, our algorithm computes in polynomial time a $(1\pm n^{-\Omega(1)})$ -approximation of the so-called ‘free energy’ of the k-colourings of G(n, d/n), for $k\geq (1+\varepsilon) d$ with probability $1-o(1)$ over the graph instances

    更新日期:2020-08-06
  • Approximately counting bases of bicircular matroids
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-08-06
    Heng Guo; Mark Jerrum

    We give a fully polynomial-time randomized approximation scheme (FPRAS) for the number of bases in bicircular matroids. This is a natural class of matroids for which counting bases exactly is #P-hard and yet approximate counting can be done efficiently.

    更新日期:2020-08-06
  • On the subgraph query problem
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-07-27
    Ryan Alweiss; Chady Ben Hamida; Xiaoyu He; Alexander Moreira

    Given a fixed graph H, a real number p ∈ (0, 1) and an infinite Erdös–Rényi graph G ∼ G(∞, p), how many adjacency queries do we have to make to find a copy of H inside G with probability at least 1/2? Determining this number f(H, p) is a variant of the subgraph query problem introduced by Ferber, Krivelevich, Sudakov and Vieira. For every graph H, we improve the trivial upper bound of f(H, p) = O(p−d)

    更新日期:2020-07-27
  • Large triangle packings and Tuza’s conjecture in sparse random graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-07-22
    Patrick Bennett; Andrzej Dudek; Shira Zerbib

    The triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle

    更新日期:2020-07-22
  • Pseudorandom hypergraph matchings
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-07-22
    Stefan Ehard; Stefan Glock; Felix Joos

    A celebrated theorem of Pippenger states that any almost regular hypergraph with small codegrees has an almost perfect matching. We show that one can find such an almost perfect matching which is ‘pseudorandom’, meaning that, for instance, the matching contains as many edges from a given set of edges as predicted by a heuristic argument.

    更新日期:2020-07-22
  • A note on distinct distances
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-07-16
    Orit E. Raz

    We show that, for a constant-degree algebraic curve γ in ℝD, every set of n points on γ spans at least Ω(n4/3) distinct distances, unless γ is an algebraic helix, in the sense of Charalambides [2]. This improves the earlier bound Ω(n5/4) of Charalambides [2]. We also show that, for every set P of n points that lie on a d-dimensional constant-degree algebraic variety V in ℝD, there exists a subset S

    更新日期:2020-07-16
  • An approximate version of Jackson’s conjecture
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-06-30
    Anita Liebenau; Yanitsa Pehova

    A diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for every ε > 0 there exists

    更新日期:2020-06-30
  • Constructing families of cospectral regular graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-06-30
    M. Haythorpe; A. Newcombe

    A set of graphs are called cospectral if their adjacency matrices have the same characteristic polynomial. In this paper we introduce a simple method for constructing infinite families of cospectral regular graphs. The construction is valid for special cases of a property introduced by Schwenk. For the case of cubic (3-regular) graphs, computational results are given which show that the construction

    更新日期:2020-06-30
  • On finite sets of small tripling or small alternation in arbitrary groups
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-06-30
    Gabriel Conant

    We prove Bogolyubov–Ruzsa-type results for finite subsets of groups with small tripling, |A3| ≤ O(|A|), or small alternation, |AA−1A| ≤ O(|A|). As applications, we obtain a qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result

    更新日期:2020-06-30
  • Ramsey properties of randomly perturbed graphs: cliques and cycles
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-06-30
    Shagnik Das; Andrew Treglown

    Given graphs H1, H2, a graph G is (H1, H2) -Ramsey if, for every colouring of the edges of G with red and blue, there is a red copy of H1 or a blue copy of H2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs. This is a random graph model introduced by Bohman, Frieze and Martin [8] in which one starts with a dense graph and then adds a given number of random

    更新日期:2020-06-30
  • Sampling biased monotonic surfaces using exponential metrics
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-06-30
    Sam Greenberg; Dana Randall; Amanda Pascoe Streib

    Monotonic surfaces spanning finite regions of ℤd arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. One method that has been used to uniformly generate these surfaces is a Markov chain that iteratively adds or removes a single cube below the surface during a step. We consider a biased version of the chain, where we are more likely to add a cube than to remove

    更新日期:2020-06-30
  • Hamiltonicity in random directed graphs is born resilient
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-06-24
    Richard Montgomery

    Let $\{D_M\}_{M\geq 0}$ be the n-vertex random directed graph process, where $D_0$ is the empty directed graph on n vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each $$\varepsilon > 0$$ , we show that, almost surely, any directed graph $D_M$ with minimum in- and out-degree at least 1 is not only Hamiltonian (as

    更新日期:2020-06-24
  • Towards the Kohayakawa–Kreuter conjecture on asymmetric Ramsey properties
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-06-24
    Frank Mousset; Rajko Nenadov; Wojciech Samotij

    For fixed graphs F1,…,Fr, we prove an upper bound on the threshold function for the property that G(n, p) → (F1,…,Fr). This establishes the 1-statement of a conjecture of Kohayakawa and Kreuter.

    更新日期:2020-06-24
  • Supersaturation of even linear cycles in linear hypergraphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-06-23
    Tao Jiang; Liana Yepremyan

    A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in an n-vertex graph not containing C2k, the cycle of length 2k, is O(n1+1/k). Simonovits established a corresponding supersaturation result for C2k’s, showing that there exist positive constants C,c depending only on K such that every n-vertex graph G with e(G) ≥ Cn1+1/k contains at least

    更新日期:2020-06-23
  • On the structure of Dense graphs with bounded clique number
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-06-05
    Heiner Oberkampf; Mathias Schacht

    We study structural properties of graphs with bounded clique number and high minimum degree. In particular, we show that there exists a function L = L(r,ɛ) such that every Kr-free graph G on n vertices with minimum degree at least ((2r–5)/(2r–3)+ɛ)n is homomorphic to a Kr-free graph on at most L vertices. It is known that the required minimum degree condition is approximately best possible for this

    更新日期:2020-06-05
  • Graph limits of random unlabelled k-trees
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-05-18
    Emma Yu Jin; Benedikt Stufler

    We study random unlabelled k-trees by combining the colouring approach by Gainer-Dewar and Gessel (2014) with the cycle-pointing method by Bodirsky, Fusy, Kang and Vigerske (2011). Our main applications are Gromov–Hausdorff–Prokhorov and Benjamini–Schramm limits that describe their asymptotic geometric shape on a global and local scale as the number of (k + 1) -cliques tends to infinity.

    更新日期:2020-05-18
  • Pointer chasing via triangular discrimination
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-05-15
    Amir Yehudayoff

    We prove an essentially sharp $\tilde \Omega (n/k)$ lower bound on the k-round distributional complexity of the k-step pointer chasing problem under the uniform distribution, when Bob speaks first. This is an improvement over Nisan and Wigderson’s $\tilde \Omega (n/{k^2})$ lower bound, and essentially matches the randomized lower bound proved by Klauck. The proof is information-theoretic, and a key

    更新日期:2020-05-15
  • Finding independent transversals efficiently
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-05-14
    Alessandra Graf; Penny Haxell

    We give an efficient algorithm that, given a graph G and a partition V1,…,Vm of its vertex set, finds either an independent transversal (an independent set {v1,…,vm} in G such that ${v_i} \in {V_i}$ for each i), or a subset ${\cal B}$ of vertex classes such that the subgraph of G induced by $\bigcup\nolimits_{\cal B}$ has a small dominating set. A non-algorithmic proof of this result has been known

    更新日期:2020-05-14
  • Estimating parameters associated with monotone properties
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-03-24
    Carlos Hoppen; Yoshiharu Kohayakawa; Richard Lang; Hanno Lefmann; Henrique Stagni

    There has been substantial interest in estimating the value of a graph parameter, i.e. of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the input. Graph parameters that may be successfully estimated in this way are said to be testable or estimable, and the sample complexity qz = qz(ε) of an estimable

    更新日期:2020-03-24
  • On minimal Ramsey graphs and Ramsey equivalence in multiple colours
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-03-09
    Dennis Clemens; Anita Liebenau; Damian Reding

    For an integer q ⩾ 2, a graph G is called q-Ramsey for a graph H if every q-colouring of the edges of G contains a monochromatic copy of H. If G is q-Ramsey for H yet no proper subgraph of G has this property, then G is called q-Ramsey-minimal for H. Generalizing a statement by Burr, Nešetřil and Rödl from 1977, we prove that, for q ⩾ 3, if G is a graph that is not q-Ramsey for some graph H, then G

    更新日期:2020-03-09
  • Percolation on an infinitely generated group
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-02-20
    Agelos Georgakopoulos; John Haslegrave

    We give an example of a long range Bernoulli percolation process on a group non-quasi-isometric with ℤ, in which clusters are almost surely finite for all values of the parameter. This random graph admits diverse equivalent definitions, and we study their ramifications. We also study its expected size and point out certain phase transitions.

    更新日期:2020-02-20
  • Analysis of non-reversible Markov chains via similarity orbits
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-02-18
    Michael C. H. Choi; Pierre Patie

    In this paper we develop an in-depth analysis of non-reversible Markov chains on denumerable state space from a similarity orbit perspective. In particular, we study the class of Markov chains whose transition kernel is in the similarity orbit of a normal transition kernel, such as that of birth–death chains or reversible Markov chains. We start by identifying a set of sufficient conditions for a Markov

    更新日期:2020-02-18
  • Turán numbers of theta graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-02-13
    Boris Bukh; Michael Tait

    The theta graph ${\Theta _{\ell ,t}}$ consists of two vertices joined by t vertex-disjoint paths, each of length $\ell $ . For fixed odd $\ell $ and large t, we show that the largest graph not containing ${\Theta _{\ell ,t}}$ has at most ${c_\ell }{t^{1 - 1/\ell }}{n^{1 + 1/\ell }}$ edges and that this is tight apart from the value of ${c_\ell }$ .

    更新日期:2020-02-13
  • On subgraphs of C2k-free graphs and a problem of Kühn and Osthus
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-02-04
    Dániel Grósz; Abhishek Methuku; Casey Tompkins

    Let c denote the largest constant such that every C6-free graph G contains a bipartite and C4-free subgraph having a fraction c of edges of G. Győri, Kensell and Tompkins showed that 3/8 ⩽ c ⩽ 2/5. We prove that c = 38. More generally, we show that for any ε > 0, and any integer k ⩾ 2, there is a C2k-free graph $G'$ which does not contain a bipartite subgraph of girth greater than 2k with more than

    更新日期:2020-02-04
  • A note on the Brown–Erdős–Sós conjecture in groups
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-02-03
    Jason Long

    We show that a dense subset of a sufficiently large group multiplication table contains either a large part of the addition table of the integers modulo some k, or the entire multiplication table of a certain large abelian group, as a subgrid. As a consequence, we show that triples systems coming from a finite group contain configurations with t triples spanning $ O(\sqrt t )$ vertices, which is the

    更新日期:2020-02-03
  • Minimax functions on Galton–Watson trees
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-12-06
    James B. Martin; Roman Stasiński

    We consider the behaviour of minimax recursions defined on random trees. Such recursions give the value of a general class of two-player combinatorial games. We examine in particular the case where the tree is given by a Galton–Watson branching process, truncated at some depth 2n, and the terminal values of the level 2n nodes are drawn independently from some common distribution. The case of a regular

    更新日期:2019-12-06
  • The replica symmetric phase of random constraint satisfaction problems
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-12-03
    Amin Coja-Oghlan; Tobias Kapetanopoulos; Noela Müller

    Random constraint satisfaction problems play an important role in computer science and combinatorics. For example, they provide challenging benchmark examples for algorithms, and they have been harnessed in probabilistic constructions of combinatorial structures with peculiar features. In an important contribution (Krzakala et al. 2007, Proc. Nat. Acad. Sci.), physicists made several predictions on

    更新日期:2019-12-03
  • Avoiding long Berge cycles: the missing cases k = r + 1 and k = r + 2
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-11-27
    Beka Ergemlidze; Ervin Győri; Abhishek Methuku; Nika Salia; Casey Tompkins; Oscar Zamora

    The maximum size of an r-uniform hypergraph without a Berge cycle of length at least k has been determined for all k ≥ r + 3 by Füredi, Kostochka and Luo and for k < r (and k = r, asymptotically) by Kostochka and Luo. In this paper we settle the remaining cases: k = r + 1 and k = r + 2, proving a conjecture of Füredi, Kostochka and Luo.

    更新日期:2019-11-27
  • Sharp concentration of the equitable chromatic number of dense random graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-11-26
    Annika Heckel

    An equitable colouring of a graph G is a vertex colouring where no two adjacent vertices are coloured the same and, additionally, the colour class sizes differ by at most 1. The equitable chromatic number χ=(G) is the minimum number of colours required for this. We study the equitable chromatic number of the dense random graph ${\mathcal{G}(n,m)}$ where $m = \left\lfloor {p\left( \matrix{ n \cr 2 \cr}

    更新日期:2019-11-26
  • Counting higher order tangencies for plane curves
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-11-26
    Joshua Zahl

    We prove that n plane algebraic curves determine O(n(k+2)/(k+1)) points of kth order tangency. This generalizes an earlier result of Ellenberg, Solymosi and Zahl on the number of (first order) tangencies determined by n plane algebraic curves.

    更新日期:2019-11-26
  • Edge-statistics on large graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-11-14
    Noga Alon; Dan Hefetz; Michael Krivelevich; Mykhaylo Tyomkyn

    The inducibility of a graph H measures the maximum number of induced copies of H a large graph G can have. Generalizing this notion, we study how many induced subgraphs of fixed order k and size ℓ a large graph G on n vertices can have. Clearly, this number is $\left( {\matrix{n \cr k}}\right)$ for every n, k and $\ell \in \left\{ {0,\left( {\matrix{k \cr 2}} \right)}\right\}$ . We conjecture that

    更新日期:2019-11-14
  • Monochromatic trees in random tournaments
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-11-07
    Matija Bucić; Sven Heberle; Shoham Letzter; Benny Sudakov

    We prove that, with high probability, in every 2-edge-colouring of the random tournament on n vertices there is a monochromatic copy of every oriented tree of order $O(n{\rm{/}}\sqrt {{\rm{log}} \ n} )$ . This generalizes a result of the first, third and fourth authors, who proved the same statement for paths, and is tight up to a constant factor.

    更新日期:2019-11-07
  • Surjectivity of near-square random matrices
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-11-06
    Hoi. H. Nguyen; Elliot Paquette

    We show that a nearly square independent and identically distributed random integral matrix is surjective over the integral lattice with very high probability. This answers a question by Koplewitz [6]. Our result extends to sparse matrices as well as to matrices of dependent entries.

    更新日期:2019-11-06
  • The string of diamonds is nearly tight for rumour spreading
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-11-04
    Omer Angel; Abbas Mehrabian; Yuval Peres

    For a rumour spreading protocol, the spread time is defined as the first time everyone learns the rumour. We compare the synchronous push&pull rumour spreading protocol with its asynchronous variant, and show that for any n-vertex graph and any starting vertex, the ratio between their expected spread times is bounded by $O({n^{1/3}}{\log ^{2/3}}n)$ . This improves the $O(\sqrt n)$ upper bound of Giakkoupis

    更新日期:2019-11-04
  • Unlabelled Gibbs partitions
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-11-04
    Benedikt Stufler

    We study random composite structures considered up to symmetry that are sampled according to weights on the inner and outer structures. This model may be viewed as an unlabelled version of Gibbs partitions and encompasses multisets of weighted combinatorial objects. We describe a general setting characterized by the formation of a giant component. The collection of small fragments is shown to converge

    更新日期:2019-11-04
  • Asymptotic Normality Through Factorial Cumulants and Partition Identities.
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2014-03-05
    Konstancja Bobecka,Paweł Hitczenko,Fernando López-Blázquez,Grzegorz Rempała,Jacek Wesołowski

    In the paper we develop an approach to asymptotic normality through factorial cumulants. Factorial cumulants arise in the same manner from factorial moments as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is a new identity for 'moments' of partitions of numbers. The general limiting result is then used to (re-)derive asymptotic normality for several models including classical

    更新日期:2019-11-01
  • On the Brownian separable permuton
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-10-24
    Mickaël Maazoun

    The Brownian separable permuton is a random probability measure on the unit square, which was introduced by Bassino, Bouvel, Féray, Gerin and Pierrot (2016) as the scaling limit of the diagram of the uniform separable permutation as size grows to infinity. We show that, almost surely, the permuton is the pushforward of the Lebesgue measure on the graph of a random measure-preserving function associated

    更新日期:2019-10-24
  • On the number of symbols that forces a transversal
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-10-21
    Peter Keevash; Liana Yepremyan

    Akbari and Alipour [1] conjectured that any Latin array of order n with at least n2/2 symbols contains a transversal. For large n, we confirm this conjecture, and moreover, we show that n399/200 symbols suffice.

    更新日期:2019-10-21
  • FKN theorem for the multislice, with applications
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-10-18
    Yuval Filmus

    The Friedgut–Kalai–Naor (FKN) theorem states that if ƒ is a Boolean function on the Boolean cube which is close to degree one, then ƒ is close to a dictator, a function depending on a single coordinate. The author has extended the theorem to the slice, the subset of the Boolean cube consisting of all vectors with fixed Hamming weight. We extend the theorem further, to the multislice, a multicoloured

    更新日期:2019-10-18
  • The Infinite limit of random permutations avoiding patterns of length three
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-10-14
    Ross G. Pinsky

    For $$\tau \in {S_3}$$ , let $$\mu _n^\tau $$ denote the uniformly random probability measure on the set of $$\tau $$ -avoiding permutations in $${S_n}$$ . Let $${\mathbb {N}^*} = {\mathbb {N}} \cup \{ \infty \} $$ with an appropriate metric and denote by $$S({\mathbb{N}},{\mathbb{N}^*})$$ the compact metric space consisting of functions $$\sigma {\rm{ = }}\{ {\sigma _i}\} _{i = 1}^\infty {\rm{ }}$$

    更新日期:2019-10-14
  • On sets of points with few odd secants
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-10-10
    Simeon Ball; Bence Csajbók

    We prove that, for q odd, a set of q + 2 points in the projective plane over the field with q elements has at least 2q − c odd secants, where c is a constant and an odd secant is a line incident with an odd number of points of the set.

    更新日期:2019-10-10
  • Minimizing the number of 5-cycles in graphs with given edge-density
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-10-09
    Patrick Bennett; Andrzej Dudek; Bernard Lidický; Oleg Pikhurko

    Motivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of the 5-cycle C5. We show that every graph of order n and size $ (1 - 1/k) \left( {\matrix{n \cr 2 }} \right) $ , where k ≥ 3 is an integer, contains at least $$({1 \over {10}} - {1 \over {2k}} + {1 \over {{k^2}}} - {1 \over {{k^3}}} + {2 \over {5{k^4}}}){n^5} + o({n^5})$$ copies of C5.

    更新日期:2019-10-09
  • Long Monotone Trails in Random Edge-Labellings of Random Graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-10-08
    Omer Angel; Asaf Ferber; Benny Sudakov; Vincent Tassion

    Given a graph G and a bijection f : E(G) → {1, 2,…,e(G)}, we say that a trail/path in G is f-increasing if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chvátal and Komlós raised the question of providing worst-case estimates of the length of the longest increasing trail/path over all edge orderings of Kn. The case of a trail was resolved by

    更新日期:2019-10-08
  • Extensions of the Erdős–Gallai theorem and Luo’s theorem
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-10-08
    Bo Ning; Xing Peng

    The famous Erdős–Gallai theorem on the Turán number of paths states that every graph with n vertices and m edges contains a path with at least (2m)/n edges. In this note, we first establish a simple but novel extension of the Erdős–Gallai theorem by proving that every graph G contains a path with at least $${{(s + 1){N_{s + 1}}(G)} \over {{N_s}(G)}} + s - 1$$ edges, where Nj(G) denotes the number of

    更新日期:2019-10-08
  • The Induced Removal Lemma in Sparse Graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-09-30
    Shachar Sapir; Asaf Shapira

    The induced removal lemma of Alon, Fischer, Krivelevich and Szegedy states that if an n-vertex graph G is ε-far from being induced H-free then G contains δH(ε) · nh induced copies of H. Improving upon the original proof, Conlon and Fox proved that 1/δH(ε)is at most a tower of height poly(1/ε), and asked if this bound can be further improved to a tower of height log(1/ε). In this paper we obtain such

    更新日期:2019-09-30
  • Expansion of Percolation Critical Points for Hamming Graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-08-05
    Lorenzo Federico; Remco Van Der Hofstad; Frank Den Hollander; Tim Hulshof

    The Hamming graph H(d, n) is the Cartesian product of d complete graphs on n vertices. Let ${m=d(n-1)}$ be the degree and $V = n^d$ be the number of vertices of H(d, n). Let $p_c^{(d)}$ be the critical point for bond percolation on H(d, n). We show that, for $d \in \mathbb{N}$ fixed and $n \to \infty$ , $$p_c^{(d)} = {1 \over m} + {{2{d^2} - 1} \over {2{{(d - 1)}^2}}}{1 \over {{m^2}}} + O({m^{ - 3}})

    更新日期:2019-08-05
  • On Komlós’ tiling theorem in random graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2019-07-25
    Rajko Nenadov; Nemanja Škorić

    Given graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}}

    更新日期:2019-07-25
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