TY - JOUR AB - For some k∈Z≥0∪{∞}, we call a linear forest k-bounded if each of its components has at most k edges. We will say a (k,ℓ)-bounded linear forest decomposition of a graph G is a partition of E(G) into the edge sets of two linear forests Fk,Fℓ where Fk is k-bounded and Fℓ is ℓ-bounded. We show that the problem of deciding whether a given graph has such a decomposition is NP-complete if both k and ℓ are at least 2, NP-complete if k≥9 and ℓ=1, and is in P for (k,ℓ)=(2,1). Before this, the only known NP-complete cases were the (2,2) and (3,3) cases. Our hardness result answers a question of Bermond et al. from 1984. We also show that planar graphs of girth at least nine decompose into a linear forest and a matching, which in particular is stronger than 3-edge-colouring such graphs. AU - Campbell, Rutger AU - Hörsch, Florian AU - Moore, Benjamin ID - 15163 IS - 6 JF - Discrete Mathematics SN - 0012-365X TI - Decompositions into two linear forests of bounded lengths VL - 347 ER - TY - JOUR AB - Let Lc,n denote the size of the longest cycle in G(n, c/n),c >1 constant. We show that there exists a continuous function f(c) such that Lc,n/n→f(c) a.s. for c>20, thus extending a result of Frieze and the author to smaller values of c. Thereafter, for c>20, we determine the limit of the probability that G(n, c/n)contains cycles of every length between the length of its shortest and its longest cycles as n→∞. AU - Anastos, Michael ID - 13042 IS - 2 JF - Electronic Journal of Combinatorics TI - A note on long cycles in sparse random graphs VL - 30 ER - TY - JOUR AB - We study multigraphs whose edge-sets are the union of three perfect matchings, M1, M2, and M3. Given such a graph G and any a1; a2; a3 2 N with a1 +a2 +a3 6 n - 2, we show there exists a matching M of G with jM \ Mij = ai for each i 2 f1; 2; 3g. The bound n - 2 in the theorem is best possible in general. We conjecture however that if G is bipartite, the same result holds with n - 2 replaced by n - 1. We give a construction that shows such a result would be tight. We also make a conjecture generalising the Ryser-Brualdi-Stein conjecture with colour multiplicities. AU - Anastos, Michael AU - Fabian, David AU - Müyesser, Alp AU - Szabó, Tibor ID - 14319 IS - 3 JF - Electronic Journal of Combinatorics TI - Splitting matchings and the Ryser-Brualdi-Stein conjecture for multisets VL - 30 ER - TY - CONF AB - We study the Hamilton cycle problem with input a random graph G ~ G(n,p) in two different settings. In the first one, G is given to us in the form of randomly ordered adjacency lists while in the second one, we are given the adjacency matrix of G. In each of the two settings we derive a deterministic algorithm that w.h.p. either finds a Hamilton cycle or returns a certificate that such a cycle does not exist for p = p(n) ≥ 0. The running times of our algorithms are O(n) and respectively, each being best possible in its own setting. AU - Anastos, Michael ID - 14344 SN - 9781611977554 T2 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms TI - Fast algorithms for solving the Hamilton cycle problem with high probability VL - 2023 ER - TY - JOUR AB - We say that (Formula presented.) if, in every edge coloring (Formula presented.), we can find either a 1-colored copy of (Formula presented.) or a 2-colored copy of (Formula presented.). The well-known states that the threshold for the property (Formula presented.) is equal to (Formula presented.), where (Formula presented.) is given by (Formula presented.) for any pair of graphs (Formula presented.) and (Formula presented.) with (Formula presented.). In this article, we show the 0-statement of the Kohayakawa–Kreuter conjecture for every pair of cycles and cliques. AU - Liebenau, Anita AU - Mattos, Letícia AU - Mendonca Dos Santos, Walner AU - Skokan, Jozef ID - 11706 IS - 4 JF - Random Structures and Algorithms SN - 1042-9832 TI - Asymmetric Ramsey properties of random graphs involving cliques and cycles VL - 62 ER - TY - JOUR AB - We prove several results about substructures in Latin squares. First, we explain how to adapt our recent work on high-girth Steiner triple systems to the setting of Latin squares, resolving a conjecture of Linial that there exist Latin squares with arbitrarily high girth. As a consequence, we see that the number of order- n Latin squares with no intercalate (i.e., no 2×2 Latin subsquare) is at least (e−9/4n−o(n))n2. Equivalently, P[N=0]≥e−n2/4−o(n2)=e−(1+o(1))EN , where N is the number of intercalates in a uniformly random order- n Latin square. In fact, extending recent work of Kwan, Sah, and Sawhney, we resolve the general large-deviation problem for intercalates in random Latin squares, up to constant factors in the exponent: for any constant 0<δ≤1 we have P[N≤(1−δ)EN]=exp(−Θ(n2)) and for any constant δ>0 we have P[N≥(1+δ)EN]=exp(−Θ(n4/3logn)). Finally, as an application of some new general tools for studying substructures in random Latin squares, we show that in almost all order- n Latin squares, the number of cuboctahedra (i.e., the number of pairs of possibly degenerate 2×2 submatrices with the same arrangement of symbols) is of order n4, which is the minimum possible. As observed by Gowers and Long, this number can be interpreted as measuring ``how associative'' the quasigroup associated with the Latin square is. AU - Kwan, Matthew Alan AU - Sah, Ashwin AU - Sawhney, Mehtaab AU - Simkin, Michael ID - 14444 IS - 2 JF - Israel Journal of Mathematics SN - 0021-2172 TI - Substructures in Latin squares VL - 256 ER - TY - JOUR AB - An n-vertex graph is called C-Ramsey if it has no clique or independent set of size Clog2n (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of ‘random-like’ properties of Ramsey graphs and the study of small-ball probability for low-degree polynomials of independent random variables. The proof proceeds via an ‘additive structure’ dichotomy on the degree sequence and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics and low-rank approximation. In particular, a key ingredient is a new sharpened version of the quadratic Carbery–Wright theorem on small-ball probability for polynomials of Gaussians, which we believe is of independent interest. One of the consequences of our result is the resolution of an old conjecture of Erdős and McKay, for which Erdős reiterated in several of his open problem collections and for which he offered one of his notorious monetary prizes. AU - Kwan, Matthew Alan AU - Sah, Ashwin AU - Sauermann, Lisa AU - Sawhney, Mehtaab ID - 14499 JF - Forum of Mathematics, Pi KW - Discrete Mathematics and Combinatorics KW - Geometry and Topology KW - Mathematical Physics KW - Statistics and Probability KW - Algebra and Number Theory KW - Analysis SN - 2050-5086 TI - Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture VL - 11 ER - TY - CONF AB - Starting with the empty graph on $[n]$, at each round, a set of $K=K(n)$ edges is presented chosen uniformly at random from the ones that have not been presented yet. We are then asked to choose at most one of the presented edges and add it to the current graph. Our goal is to construct a Hamiltonian graph with $(1+o(1))n$ edges within as few rounds as possible. We show that in this process, one can build a Hamiltonian graph of size $(1+o(1))n$ in $(1+o(1))(1+(\log n)/2K) n$ rounds w.h.p. The case $K=1$ implies that w.h.p. one can build a Hamiltonian graph by choosing $(1+o(1))n$ edges in an online fashion as they appear along the first $(0.5+o(1))n\log n$ rounds of the random graph process. This answers a question of Frieze, Krivelevich and Michaeli. Observe that the number of rounds is asymptotically optimal as the first $0.5n\log n$ edges do not span a Hamilton cycle w.h.p. The case $K=\Theta(\log n)$ implies that the Hamiltonicity threshold of the corresponding Achlioptas process is at most $(1+o(1))(1+(\log n)/2K) n$. This matches the $(1-o(1))(1+(\log n)/2K) n$ lower bound due to Krivelevich, Lubetzky and Sudakov and resolves the problem of determining the Hamiltonicity threshold of the Achlioptas process with $K=\Theta(\log n)$. We also show that in the above process one can construct a graph $G$ that spans a matching of size $\lfloor V(G)/2) \rfloor$ and $(0.5+o(1))n$ edges within $(1+o(1))(0.5+(\log n)/2K) n$ rounds w.h.p. Our proof relies on a robust Hamiltonicity property of the strong $4$-core of the binomial random graph which we use as a black-box. This property allows it to absorb paths covering vertices outside the strong $4$-core into a cycle. AU - Anastos, Michael ID - 14867 T2 - Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications TI - Constructing Hamilton cycles and perfect matchings efficiently ER - TY - JOUR AB - We show that the number of linear spaces on a set of n points and the number of rank-3 matroids on a ground set of size n are both of the form (cn+o(n))n2/6, where c=e3√/2−3(1+3–√)/2. This is the final piece of the puzzle for enumerating fixed-rank matroids at this level of accuracy: the numbers of rank-1 and rank-2 matroids on a ground set of size n have exact representations in terms of well-known combinatorial functions, and it was recently proved by van der Hofstad, Pendavingh, and van der Pol that for constant r≥4 there are (e1−rn+o(n))nr−1/r! rank-r matroids on a ground set of size n. In our proof, we introduce a new approach for bounding the number of clique decompositions of a complete graph, using quasirandomness instead of the so-called entropy method that is common in this area. AU - Kwan, Matthew Alan AU - Sah, Ashwin AU - Sawhney, Mehtaab ID - 15173 IS - G2 JF - Comptes Rendus Mathematique SN - 1631-073X TI - Enumerating matroids and linear spaces VL - 361 ER - TY - JOUR AB - In this note, we study large deviations of the number 𝐍 of intercalates ( 2×2 combinatorial subsquares which are themselves Latin squares) in a random 𝑛×𝑛 Latin square. In particular, for constant 𝛿>0 we prove that exp(−𝑂(𝑛2log𝑛))⩽Pr(𝐍⩽(1−𝛿)𝑛2/4)⩽exp(−Ω(𝑛2)) and exp(−𝑂(𝑛4/3(log𝑛)))⩽Pr(𝐍⩾(1+𝛿)𝑛2/4)⩽exp(−Ω(𝑛4/3(log𝑛)2/3)) . As a consequence, we deduce that a typical order- 𝑛 Latin square has (1+𝑜(1))𝑛2/4 intercalates, matching a lower bound due to Kwan and Sudakov and resolving an old conjecture of McKay and Wanless. AU - Kwan, Matthew Alan AU - Sah, Ashwin AU - Sawhney, Mehtaab ID - 11186 IS - 4 JF - Bulletin of the London Mathematical Society SN - 0024-6093 TI - Large deviations in random latin squares VL - 54 ER - TY - CONF AB - List-decodability of Reed-Solomon codes has re-ceived a lot of attention, but the best-possible dependence between the parameters is still not well-understood. In this work, we focus on the case where the list-decoding radius is of the form r=1−ε for ε tending to zero. Our main result states that there exist Reed-Solomon codes with rate Ω(ε) which are (1−ε,O(1/ε) -list-decodable, meaning that any Hamming ball of radius 1−ε contains at most O(1/ε) codewords. This trade-off between rate and list-decoding radius is best-possible for any code with list size less than exponential in the block length. By achieving this trade-off between rate and list-decoding radius we improve a recent result of Guo, Li, Shangguan, Tamo, and Wootters, and resolve the main motivating question of their work. Moreover, while their result requires the field to be exponentially large in the block length, we only need the field size to be polynomially large (and in fact, almost-linear suffices). We deduce our main result from a more general theorem, in which we prove good list-decodability properties of random puncturings of any given code with very large distance. AU - Ferber, Asaf AU - Kwan, Matthew Alan AU - Sauermann, Lisa ID - 11145 SN - 0272-5428 T2 - 62nd Annual IEEE Symposium on Foundations of Computer Science TI - List-decodability with large radius for Reed-Solomon codes VL - 2022 ER - TY - JOUR AB - List-decodability of Reed–Solomon codes has received a lot of attention, but the best-possible dependence between the parameters is still not well-understood. In this work, we focus on the case where the list-decoding radius is of the form r = 1-ε for ε tending to zero. Our main result states that there exist Reed–Solomon codes with rate Ω(ε) which are (1 - ε, O(1/ε))-list-decodable, meaning that any Hamming ball of radius 1-ε contains at most O(1/ε) codewords. This trade-off between rate and list-decoding radius is best-possible for any code with list size less than exponential in the block length. By achieving this trade-off between rate and list-decoding radius we improve a recent result of Guo, Li, Shangguan, Tamo, and Wootters, and resolve the main motivating question of their work. Moreover, while their result requires the field to be exponentially large in the block length, we only need the field size to be polynomially large (and in fact, almost-linear suffices). We deduce our main result from a more general theorem, in which we prove good list-decodability properties of random puncturings of any given code with very large distance. AU - Ferber, Asaf AU - Kwan, Matthew Alan AU - Sauermann, Lisa ID - 10775 IS - 6 JF - IEEE Transactions on Information Theory SN - 0018-9448 TI - List-decodability with large radius for Reed-Solomon codes VL - 68 ER - TY - JOUR AB - Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope P is defined to be the minimum number of facets of a (possibly higher-dimensional) polytope from which P can be obtained as a (linear) projection. This notion is motivated by its relevance to combinatorial optimisation, and has been studied intensively for various specific polytopes associated with important optimisation problems. In this paper we study extension complexity as a parameter of general polytopes, more specifically considering various families of low-dimensional polytopes. First, we prove that for a fixed dimension d, the extension complexity of a random d-dimensional polytope (obtained as the convex hull of random points in a ball or on a sphere) is typically on the order of the square root of its number of vertices. Second, we prove that any cyclic n-vertex polygon (whose vertices lie on a circle) has extension complexity at most 24√n. This bound is tight up to the constant factor 24. Finally, we show that there exists an no(1)-dimensional polytope with at most n vertices and extension complexity n1−o(1). Our theorems are proved with a range of different techniques, which we hope will be of further interest. AU - Kwan, Matthew Alan AU - Sauermann, Lisa AU - Zhao, Yufei ID - 11443 IS - 6 JF - Transactions of the American Mathematical Society SN - 0002-9947 TI - Extension complexity of low-dimensional polytopes VL - 375 ER - TY - JOUR AB - We consider a generalised model of a random simplicial complex, which arises from a random hypergraph. Our model is generated by taking the downward-closure of a non-uniform binomial random hypergraph, in which for each k, each set of k+1 vertices forms an edge with some probability pk independently. As a special case, this contains an extensively studied model of a (uniform) random simplicial complex, introduced by Meshulam and Wallach [Random Structures & Algorithms 34 (2009), no. 3, pp. 408–417]. We consider a higher-dimensional notion of connectedness on this new model according to the vanishing of cohomology groups over an arbitrary abelian group R. We prove that this notion of connectedness displays a phase transition and determine the threshold. We also prove a hitting time result for a natural process interpretation, in which simplices and their downward-closure are added one by one. In addition, we determine the asymptotic behaviour of cohomology groups inside the critical window around the time of the phase transition. AU - Cooley, Oliver AU - Del Giudice, Nicola AU - Kang, Mihyun AU - Sprüssel, Philipp ID - 11740 IS - 3 JF - Electronic Journal of Combinatorics TI - Phase transition in cohomology groups of non-uniform random simplicial complexes VL - 29 ER - TY - JOUR AB - The k-sample G(k,W) from a graphon W:[0,1]2→[0,1] is the random graph on {1,…,k}, where we sample x1,…,xk∈[0,1] uniformly at random and make each pair {i,j}⊆{1,…,k} an edge with probability W(xi,xj), with all these choices being mutually independent. Let the random variable Xk(W) be the number of edges in G(k,W). Vera T. Sós asked in 2012 whether two graphons U, W are necessarily weakly isomorphic if the random variables Xk(U) and Xk(W) have the same distribution for every integer k≥2. This question when one of the graphons W is a constant function was answered positively by Endre Csóka and independently by Jacob Fox, Tomasz Łuczak and Vera T. Sós. Here we investigate the question when W is a 2-step graphon and prove that the answer is positive for a 3-dimensional family of such graphons. We also present some related results. AU - Cooley, Oliver AU - Kang, M. AU - Pikhurko, O. ID - 12151 JF - Acta Mathematica Hungarica KW - graphon KW - k-sample KW - graphon forcing KW - graph container SN - 0236-5294 TI - On a question of Vera T. Sós about size forcing of graphons VL - 168 ER - TY - CONF AB - We present CertifyHAM, a deterministic algorithm that takes a graph G as input and either finds a Hamilton cycle of G or outputs that such a cycle does not exist. If G ∼ G(n, p) and p ≥ 100 log n/n then the expected running time of CertifyHAM is O(n/p) which is best possible. This improves upon previous results due to Gurevich and Shelah, Thomason and Alon, and Krivelevich, who proved analogous results for p being constant, p ≥ 12n −1/3 and p ≥ 70n −1/2 respectively. AU - Anastos, Michael ID - 12432 SN - 0272-5428 T2 - 63rd Annual IEEE Symposium on Foundations of Computer Science TI - Solving the Hamilton cycle problem fast on average VL - 2022-October ER - TY - JOUR AB - Inspired by the study of loose cycles in hypergraphs, we define the loose core in hypergraphs as a structurewhich mirrors the close relationship between cycles and $2$-cores in graphs. We prove that in the $r$-uniform binomial random hypergraph $H^r(n,p)$, the order of the loose core undergoes a phase transition at a certain critical threshold and determine this order, as well as the number of edges, asymptotically in the subcritical and supercritical regimes. Our main tool is an algorithm called CoreConstruct, which enables us to analyse a peeling process for the loose core. By analysing this algorithm we determine the asymptotic degree distribution of vertices in the loose core and in particular how many vertices and edges the loose core contains. As a corollary we obtain an improved upper bound on the length of the longest loose cycle in $H^r(n,p)$. AU - Cooley, Oliver AU - Kang, Mihyun AU - Zalla, Julian ID - 12286 IS - 4 JF - The Electronic Journal of Combinatorics KW - Computational Theory and Mathematics KW - Geometry and Topology KW - Theoretical Computer Science KW - Applied Mathematics KW - Discrete Mathematics and Combinatorics TI - Loose cores and cycles in random hypergraphs VL - 29 ER -