• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-10-27

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
• 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
• 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
• 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
• 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
• 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 $$$$\gamma(G) \leq \frac{n}{d}(\log d + 1).$$$$ In this paper the main result is that if G

更新日期：2020-10-16
• 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
• 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
• Comb. Probab. Comput. (IF 0.879) Pub Date : 2020-10-12

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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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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