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  • The length of an s-increasing sequence of r-tuples
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2021-01-08
    W. T. Gowers; J. Long

    We prove a number of results related to a problem of Po-Shen Loh [9], which is equivalent to a problem in Ramsey theory. Let a = (a1, a2, a3) and b = (b1, b2, b3) be two triples of integers. Define a to be 2-less than b if ai < bi for at least two values of i, and define a sequence a1, …, am of triples to be 2-increasing if ar is 2-less than as whenever r < s. Loh asks how long a 2-increasing sequence

    更新日期:2021-01-08
  • Cycle partitions of regular graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-12-18
    Vytautas Gruslys; Shoham Letzter

    Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths

    更新日期:2020-12-18
  • Counting Hamilton cycles in Dirac hypergraphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-12-17
    Stefan Glock; Stephen Gould; Felix Joos; Daniela Kühn; Deryk Osthus

    A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rödl, Ruciński and Szemerédi proved that for $k\ge 3$ , every k-graph on n vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs

    更新日期:2020-12-18
  • Near-perfect clique-factors in sparse pseudorandom graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-12-11
    Jie Han; Yoshiharu Kohayakawa; Yury Person

    We prove that, for any $t \ge 3$, there exists a constant c = c(t) > 0 such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying $\lambda \le c{d^{t - 1}}/{n^{t - 2}}$ contains vertex-disjoint copies of kt covering all but at most ${n^{1 - 1/(8{t^4})}}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo (Combinatorica

    更新日期:2020-12-18
  • Large complete minors in random subgraphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-12-03
    Joshua Erde; Mihyun Kang; Michael Krivelevich

    Let G be a graph of minimum degree at least k and let Gp be the random subgraph of G obtained by keeping each edge independently with probability p. We are interested in the size of the largest complete minor that Gp contains when p = (1 + ε)/k with ε > 0. We show that with high probability Gp contains a complete minor of order $\tilde{\Omega}(\sqrt{k})$ , where the ~ hides a polylogarithmic factor

    更新日期:2020-12-18
  • Generalizations of the Ruzsa–Szemerédi and rainbow Turán problems for cliques
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-11-19
    W. T. Gowers; Barnabás Janzer

    Considering a natural generalization of the Ruzsa–Szemerédi problem, we prove that for any fixed positive integers r, s with r < s, there are graphs on n vertices containing $n^{r}e^{-O\left(\sqrt{\log{n}}\right)}=n^{r-o(1)}$ copies of Ks such that any Kr is contained in at most one Ks. We also give bounds for the generalized rainbow Turán problem ex (n, H, rainbow - F) when F is complete. In particular

    更新日期:2020-11-19
  • Lipschitz bijections between boolean functions
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-11-16
    Tom Johnston; Alex Scott

    We answer four questions from a recent paper of Rao and Shinkar [17] on Lipschitz bijections between functions from {0, 1}n to {0, 1}. (1) We show that there is no O(1)-bi-Lipschitz bijection from Dictator to XOR such that each output bit depends on O(1) input bits. (2) We give a construction for a mapping from XOR to Majority which has average stretch $O(\sqrt{n})$ , matching a previously known lower

    更新日期:2020-11-16
  • Making Kr+1-free graphs r-partite
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-11-10
    József Balogh; Felix Christian Clemen; Mikhail Lavrov; Bernard Lidický; Florian Pfender

    The Erdős–Simonovits stability theorem states that for all ε > 0 there exists α > 0 such that if G is a Kr+1-free graph on n vertices with e(G) > ex(n, Kr+1)– α n2, then one can remove εn2 edges from G to obtain an r-partite graph. Füredi gave a short proof that one can choose α = ε. We give a bound for the relationship of α and ε which is asymptotically sharp as ε → 0.

    更新日期:2020-11-12
  • Subgraph counts for dense random graphs with specified degrees
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-11-05
    Catherine Greenhill; Mikhail Isaev; Brendan D. McKay

    We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence(d 1 , …, d n ) as n→ ∞. We also determine the expected number of spanning trees in this model. The range of degrees

    更新日期:2020-11-05
  • Tail bounds on hitting times of randomized search heuristics using variable drift analysis
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-11-05
    P. K. Lehre; C. Witt

    Drift analysis is one of the state-of-the-art techniques for the runtime analysis of randomized search heuristics (RSHs) such as evolutionary algorithms (EAs), simulated annealing, etc. The vast majority of existing drift theorems yield bounds on the expected value of the hitting time for a target state, for example the set of optimal solutions, without making additional statements on the distribution

    更新日期:2020-11-05
  • Disjointness graphs of segments in the space
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-11-04
    János Pach; Gábor Tardos; Géza Tóth

    The disjointness graph G = G(𝒮) of a set of segments 𝒮 in ${\mathbb{R}^d}$ , $$d \ge 2$$ , is a graph whose vertex set is 𝒮 and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies $\chi (G) \le {(\omega (G))^4} + {(\omega (G))^3}$ , where ω(G) denotes the clique number of G. It follows that 𝒮 has Ω(n1/5)

    更新日期:2020-11-04
  • A discrepancy version of the Hajnal–Szemerédi theorem
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-10-30
    József Balogh; Béla Csaba; András Pluhár; Andrew Treglown

    A perfect Kr-tiling in a graph G is a collection of vertex-disjoint copies of the clique Kr in G covering every vertex of G. The famous Hajnal–Szemerédi theorem determines the minimum degree threshold for forcing a perfect Kr-tiling in a graph G. The notion of discrepancy appears in many branches of mathematics. In the graph setting, one assigns the edges of a graph G labels from {‒1, 1}, and one seeks

    更新日期:2020-10-30
  • The minimum perfect matching in pseudo-dimension 0 < q < 1
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-10-27
    Joel Larsson

    It is known that for Kn,n equipped with i.i.d. exp (1) edge costs, the minimum total cost of a perfect matching converges to $\zeta(2)=\pi^2/6$ in probability. Similar convergence has been established for all edge cost distributions of pseudo-dimension $q \geq 1$ . In this paper we extend those results to all real positive q, confirming the Mézard–Parisi conjecture in the last remaining applicable

    更新日期:2020-10-30
  • A simplified disproof of Beck’s three permutations conjecture and an application to root-mean-squared discrepancy
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-10-26
    Cole Franks

    A k-permutation family on n vertices is a set-system consisting of the intervals of k permutations of the integers 1 to n. The discrepancy of a set-system is the minimum over all red–blue vertex colourings of the maximum difference between the number of red and blue vertices in any set in the system. In 2011, Newman and Nikolov disproved a conjecture of Beck that the discrepancy of any 3-permutation

    更新日期:2020-10-30
  • Frozen (Δ + 1)-colourings of bounded degree graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-10-19
    Marthe Bonamy; Nicolas Bousquet; Guillem Perarnau

    Let G be a graph on n vertices and with maximum degree Δ, and let k be an integer. The k-recolouring graph of G is the graph whose vertices are k-colourings of G and where two k-colourings are adjacent if they differ at exactly one vertex. It is well known that the k-recolouring graph is connected for $k\geq \Delta+2$ . Feghali, Johnson and Paulusma (J. Graph Theory 83 (2016) 340–358) showed that the

    更新日期:2020-10-19
  • Mixing properties of colourings of the ℤd lattice
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-10-19
    Noga Alon; Raimundo Briceño; Nishant Chandgotia; Alexander Magazinov; Yinon Spinka

    We study and classify proper q-colourings of the ℤd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When $q\le d+1$ , there exist frozen colourings, that is, proper q-colourings of ℤd which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when $q\ge d+2$ , any proper q-colouring of the boundary of a box

    更新日期:2020-10-19
  • A non-increasing tree growth process for recursive trees and applications
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-10-19
    Laura Eslava

    We introduce a non-increasing tree growth process $((T_n,{\sigma}_n),\, n\ge 1)$ , where Tn is a rooted labelled tree on n vertices and σn is a permutation of the vertex labels. The construction of (Tn, σn) from (Tn−1, σn−1) involves rewiring a random (possibly empty) subset of edges in Tn−1 towards the newly added vertex; as a consequence Tn−1 ⊄ Tn with positive probability. The key feature of the

    更新日期:2020-10-19
  • Independent dominating sets in graphs of girth five
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-10-15
    Ararat Harutyunyan; Paul Horn; Jacques Verstraete

    Let $\gamma(G)$ and $${\gamma _ \circ }(G)$$ denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n-vertex graph of minimum degree at least d, then $$\begin{equation}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{equation}$$ In this paper the main result is that if G

    更新日期:2020-10-16
  • Covering and tiling hypergraphs with tight cycles
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-10-13
    Jie Han; Allan Lo; Nicolás Sanhueza-Matamala

    A k-uniform tight cycle $C_s^k$ is a hypergraph on s > k vertices with a cyclic ordering such that every k consecutive vertices under this ordering form an edge. The pair (k, s) is admissible if gcd (k, s) = 1 or k / gcd (k,s) is even. We prove that if $s \ge 2{k^2}$ and H is a k-uniform hypergraph with minimum codegree at least (1/2 + o(1))|V(H)|, then every vertex is covered by a copy of $C_s^k$

    更新日期:2020-10-13
  • Sharp bounds for decomposing graphs into edges and triangles
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-10-12
    Adam Blumenthal; Bernard Lidický; Yanitsa Pehova; Florian Pfender; Oleg Pikhurko; Jan Volec

    For a real constant α, let $\pi _3^\alpha (G)$ be the minimum of twice the number of K2’s plus α times the number of K3’s over all edge decompositions of G into copies of K2 and K3, where Kr denotes the complete graph on r vertices. Let $\pi _3^\alpha (n)$ be the maximum of $\pi _3^\alpha (G)$ over all graphs G with n vertices. The extremal function $\pi _3^3(n)$ was first studied by Győri and Tuza

    更新日期:2020-10-12
  • Tight Hamilton cycles in cherry-quasirandom 3-uniform hypergraphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-10-12
    Elad Aigner-Horev; Gil Levy

    We employ the absorbing-path method in order to prove two results regarding the emergence of tight Hamilton cycles in the so-called two-path or cherry-quasirandom 3-graphs. Our first result asserts that for any fixed real α > 0, cherry-quasirandom 3-graphs of sufficiently large order n having minimum 2-degree at least α(n – 2) have a tight Hamilton cycle. Our second result concerns the minimum 1-degree

    更新日期:2020-10-12
  • Eigenvalues and triangles in graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-09-28
    Huiqiu Lin; Bo Ning; Baoyindureng Wu

    Bollobás and Nikiforov (J. Combin. Theory Ser. B.97 (2007) 859–865) conjectured the following. If G is a Kr+1-free graph on at least r+1 vertices and m edges, then ${\rm{\lambda }}_1^2(G) + {\rm{\lambda }}_2^2(G) \le (r - 1)/r \cdot 2m$ , where λ1 (G)and λ2 (G) are the largest and the second largest eigenvalues of the adjacency matrix A(G), respectively. In this paper we confirm the conjecture in the

    更新日期:2020-09-28
  • Finding tight Hamilton cycles in random hypergraphs faster
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-09-23
    Peter Allen; Christoph Koch; Olaf Parczyk; Yury Person

    In an r-uniform hypergraph on n vertices, a tight Hamilton cycle consists of n edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of r vertices. We provide a first deterministic polynomial-time algorithm, which finds a.a.s. tight Hamilton cycles in random r-uniform hypergraphs with edge probability at least C log3n/n. Our result partially

    更新日期:2020-09-23
  • Maker–Breaker percolation games I: crossing grids
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-09-15
    A. Nicholas Day; Victor Falgas-Ravry

    Motivated by problems in percolation theory, we study the following two-player positional game. Let Λm×n be a rectangular grid-graph with m vertices in each row and n vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims p (as yet unclaimed) edges of the board Λm×n, while on each of his turns Breaker claims q (as yet unclaimed) edges

    更新日期:2020-09-15
  • 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:2020-09-14
  • Improved Ramsey-type results for comparability graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-08-13
    Dániel Korándi; István Tomon

    Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if G is an n-vertex graph that is the union of r comparability (or more generally, perfect) graphs, then either G or its complement contains a clique of size $n^{1/(r+1)}$ . This

    更新日期:2020-09-14
  • 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:2020-09-14
  • Triangle-degrees in graphs and tetrahedron coverings in 3-graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-09-09
    Victor Falgas-Ravry; Klas Markström; Yi Zhao

    We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F, what is c1(n, F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex-degree $\delta_1(G)>d$ then every vertex of G is contained in a copy of F in G? We asymptotically determine c1(n, F) when F is the generalized triangle $K_4^{(3)-}$ , and we give close to optimal bounds in the case

    更新日期:2020-09-10
  • Hamiltonian Berge cycles in random hypergraphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-09-08
    Deepak Bal; Ross Berkowitz; Pat Devlin; Mathias Schacht

    In this note we study the emergence of Hamiltonian Berge cycles in random r-uniform hypergraphs. For $r\geq 3$ we prove an optimal stopping time result that if edges are sequentially added to an initially empty r-graph, then as soon as the minimum degree is at least 2, the hypergraph with high probability has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity

    更新日期:2020-09-08
  • Dirac’s theorem for random regular graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-08-28
    Padraig Condon; Alberto Espuny Díaz; António Girão; Daniela Kühn; Deryk Osthus

    We prove a ‘resilience’ version of Dirac’s theorem in the setting of random regular graphs. More precisely, we show that whenever d is sufficiently large compared to $\epsilon > 0$ , a.a.s. the following holds. Let $G'$ be any subgraph of the random n-vertex d-regular graph $G_{n,d}$ with minimum degree at least $$(1/2 + \epsilon )d$$ . Then $G'$ is Hamiltonian. This proves a conjecture of Ben-Shimon

    更新日期:2020-08-28
  • Resilience of the rank of random matrices
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-08-28
    Asaf Ferber; Kyle Luh; Gweneth McKinley

    Let M be an n × m matrix of independent Rademacher (±1) random variables. It is well known that if $n \leq m$ , then M is of full rank with high probability. We show that this property is resilient to adversarial changes to M. More precisely, if $m \ge n + {n^{1 - \varepsilon /6}}$ , then even after changing the sign of (1 – ε)m/2 entries, M is still of full rank with high probability. Note that this

    更新日期:2020-08-28
  • Breaking bivariate records
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-08-18
    James Allen Fill

    We establish a fundamental property of bivariate Pareto records for independent observations uniformly distributed in the unit square. We prove that the asymptotic conditional distribution of the number of records broken by an observation given that the observation sets a record is Geometric with parameter 1/2.

    更新日期:2020-08-18
  • Many disjoint triangles in co-triangle-free graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-08-14
    Mykhaylo Tyomkyn

    We prove that any n-vertex graph whose complement is triangle-free contains n2/12 – o(n2) edge-disjoint triangles. This is tight for the disjoint union of two cliques of order n/2. We also prove a corresponding stability theorem, that all large graphs attaining the above bound are close to being bipartite. Our results answer a question of Alon and Linial, and make progress on a conjecture of Erdős

    更新日期:2020-08-14
  • Monochromatic cycle partitions in random graphs
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-08-14
    Richard Lang; Allan Lo

    Erdős, Gyárfás and Pyber showed that every r-edge-coloured complete graph Kn can be covered by 25 r2 log r vertex-disjoint monochromatic cycles (independent of n). Here we extend their result to the setting of binomial random graphs. That is, we show that if $p = p(n) = \Omega(n^{-1/(2r)})$ , then with high probability any r-edge-coloured G(n, p) can be covered by at most 1000r4 log r vertex-disjoint

    更新日期:2020-08-14
  • Robustness of randomized rumour spreading
    Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-08-12
    Rami Daknama; Konstantinos Panagiotou; Simon Reisser

    In this work we consider three well-studied broadcast protocols: push, pull and push&pull. A key property of all these models, which is also an important reason for their popularity, is that they are presumed to be very robust, since they are simple, randomized and, crucially, do not utilize explicitly the global structure of the underlying graph. While sporadic results exist, there has been no systematic

    更新日期:2020-08-12
  • 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:2020-08-07
  • 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
  • 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
  • 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, |A 3| ≤ 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 F 1,…,F r , we prove an upper bound on the threshold function for the property that G(n, p) → (F 1,…,F r ). 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
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