@article{21159, abstract = {One of the foundational theorems of extremal graph theory is Dirac’s theorem, which says that if an n-vertex graph G has minimum degree at least n/2, then G has a Hamilton cycle, and therefore a perfect matching (if n is even). Later work by Sárközy, Selkow and Szemerédi showed that in fact Dirac graphs have many Hamilton cycles and perfect matchings, culminating in a result of Cuckler and Kahn that gives a precise description of the numbers of Hamilton cycles and perfect matchings in a Dirac graph G (in terms of an entropy-like parameter of G). In this paper we extend Cuckler and Kahn’s result to perfect matchings in hypergraphs. For positive integers d < k, and for n divisible by k, let md (k, n) be the minimum d-degree that ensures the existence of a perfect matching in an n-vertex k-uniform hypergraph. In general, it is an open question to determine (even asymptotically) the values of md (k, n), but we are nonetheless able to prove an analogue of the Cuckler–Kahn theorem, showing that if an n-vertex k-uniform hypergraph G has minimum d-degree at least (1+γ )md (k, n) (for any constantγ > 0), then the number of perfect matchings in G is controlled by an entropy-like parameter of G. This strengthens cruder estimates arising from work of Kang–Kelly–Kühn–Osthus–Pfenninger and Pham–Sah–Sawhney–Simkin.}, author = {Kwan, Matthew Alan and Safavi Hemami, Roodabeh and Wang, Yiting}, issn = {1439-6912}, journal = {Combinatorica}, publisher = {Springer Nature}, title = {{Counting perfect matchings in Dirac hypergraphs}}, doi = {10.1007/s00493-025-00194-8}, volume = {46}, year = {2026}, } @article{21884, abstract = {We show that a randomly perturbed digraph, where we start with a dense digraph Dα and add a small number of random edges to it, will typically contain a fixed orientation of a bounded-degree spanning tree. This answers a question posed by Araujo, Balogh, Krueger, Piga and Treglown and generalizes the corresponding result for randomly perturbed graphs by Krivelevich, Kwan and Sudakov. More specifically, we prove that there exists a constant c=c(α,Δ) such that if T is an oriented tree with maximum degree Δ and Dα is an n-vertex digraph with minimum semidegree αn, then the graph obtained by adding cn uniformly random edges to Dα will contain T with high probability.}, author = {Morawski, Patryk and Petrova, Kalina H}, issn = {1077-8926}, journal = {Electronic Journal of Combinatorics}, number = {2}, publisher = {Electronic Journal of Combinatorics}, title = {{Randomly perturbed digraphs also have bounded-degree spanning trees}}, doi = {10.37236/13316}, volume = {33}, year = {2026}, } @article{20422, abstract = {We show that if n is odd and p>=Clog n/n, then with high probability Hamilton cycles in G(n,p) span its cycle space. More generally, we show this holds for a class of graphs satisfying certain natural pseudorandom properties. The proof is based on a novel idea of parity-switchers, which can be thought of as analogues of absorbers in the context of cycle spaces. As another application of our method, we show that Hamilton cycles in a near-Dirac graph G, that is, a graph G with odd n vertices and minimum degree n/2+C for sufficiently large constant C, span its cycle space. }, author = {Christoph, Micha and Nenadov, Rajko and Petrova, Kalina H}, issn = {1096-0902}, journal = {Journal of Combinatorial Theory Series B}, pages = {254--267}, publisher = {Elsevier}, title = {{The Hamilton space of pseudorandom graphs}}, doi = {10.1016/j.jctb.2025.09.002}, volume = {176}, year = {2026}, } @article{20482, abstract = {In his study of graph codes, Alon introduced the concept of the odd-Ramsey number of a family of graphs H in Kn, defined as the minimum number of colours needed to colour the edges of K so that every copy of a graph H E H intersects some colour class in an odd number of edges. In this paper, we focus on complete bipartite graphs. First, we completely resolve the problem when H is the family of all spanning complete bipartite graphs on n vertices. We then focus on its subfamilies, that is, {Kt,n-t : t E T} for a fixed set of integers T c [[n/2]]. We prove that the odd-Ramsey problem is equivalent to determining the maximum dimension of a linear binary code avoiding codewords of given weights, and leverage known results from coding theory to deduce asymptotically tight bounds in our setting. We conclude with bounds for the odd-Ramsey numbers of fixed (that is, non-spanning) complete bipartite subgraphs.}, author = {Boyadzhiyska, Simona and Das, Shagnik and Lesgourgues, Thomas and Petrova, Kalina H}, issn = {0195-6698}, journal = {European Journal of Combinatorics}, publisher = {Elsevier}, title = {{Odd-Ramsey numbers of complete bipartite graphs}}, doi = {10.1016/j.ejc.2025.104235}, volume = {131}, year = {2026}, } @unpublished{21211, abstract = {We show that, for any graph F and η > 0, there exists a d0 = d0(F, η) such that every nvertex d-regular graph with d ≥ d0 has a collection of vertex-disjoint F-subdivisions covering at least (1 − η)n vertices. This verifies a conjecture of Verstraëte from 2002 and improves a recent result of Letzter, Methuku and Sudakov which additionally required d to be at least polylogarithmic in n. }, author = {Montgomery, Richard and Petrova, Kalina H and Ranganathan, Arjun and Tan, Jane}, booktitle = {arXiv}, title = {{Packing subdivisions into regular graphs}}, doi = {10.48550/arXiv.2508.00480}, year = {2025}, } @article{21263, abstract = {Two landmark results in combinatorial random matrix theory, due to Komlós and Costello–Tao–Vu, show that discrete random matrices and symmetric discrete random matrices are typically nonsingular. In particular, in the language of graph theory, when p is a fixed constant, the biadjacency matrix of a random Erdős–Rényi bipartite graph G(n,n,p) and the adjacency matrix of an Erdős–Rényi random graph G(n,p) are both nonsingular with high probability. However, very sparse random graphs (i.e., where p is allowed to decay rapidly with n) are typically singular, due to the presence of “local” dependencies such as isolated vertices and pairs of degree-1 vertices with the same neighbour. In this paper, we give a combinatorial description of the rank of a sparse random graph G(n,n,c/n) or G(n,c/n) in terms of such local dependencies, for all constants c=e (and we present some evidence that the situation is very different for c=e). This gives an essentially complete answer to a question raised by Vu (2014). As applications of our main theorem and its proof, we also determine the asymptotic singularity probability of the 2-core of a sparse random graph, we show that the rank of a sparse random graph is extremely well approximated by its matching number, and we deduce a central limit theorem for the rank of G(n,c/n).}, author = {Glasgow, Margalit and Kwan, Matthew Alan and Sah, Ashwin and Sawhney, Mehtaab}, issn = {1435-9863}, journal = {Journal of the European Mathematical Society}, publisher = {European Mathematical Society Press}, title = {{The exact rank of sparse random graphs}}, doi = {10.4171/jems/1692}, year = {2025}, } @article{21706, abstract = {Consider a quadratic polynomial Q(ξ1, . . . , ξn) of independent Rademacher random variables ξ1, . . . , ξn. To what extent can Q(ξ1, . . . , ξn) concentrate on a single value? This quadratic version of the classical Littlewood–Offord problem was popularised by Costello, Tao and Vu in their study of symmetric random matrices. In this paper, we obtain an essentially optimal bound for this problem, as conjectured by Nguyen and Vu. Specifically, if Q(ξ1, . . . , ξn) ‘robustly depends on at least m of the ξi’ in the sense that there is no way to pin down the value of Q(ξ1, . . . , ξn) by fixing values for fewer than m of the variables ξi, then we have Pr[Q(ξ1, . . . , ξn) = 0] ≤ O(1/√m). This also implies a similar result in the case where ξ1, . . . , ξn have arbitrary distributions. Our proof combines a number of ideas that may be of independent interest, including an inductive decoupling scheme that reduces quadratic anticoncentration problems to high-dimensional linear anticoncentration problems. Also, one application of our main result is the resolution of a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn related to graph inducibility. }, author = {Kwan, Matthew Alan and Sauermann, Lisa}, issn = {1570-5846}, journal = {Compositio Mathematica}, number = {12}, pages = {3089--3139}, publisher = {Cambridge University Press}, title = {{Resolution of the quadratic Littlewood–Offord problem}}, doi = {10.1112/S0010437X25102789}, volume = {161}, year = {2025}, } @article{18478, abstract = {For a given graph G=(V,E), we define its \emph{nth subdivision} as the graph obtained from G by replacing every edge by a path of length n. We also define the \emph{mth power} of G as the graph on vertex set V where we connect every pair of vertices at distance at most m in G. In this paper, we study the chromatic number of powers of subdivisions of graphs and resolve the case m=n asymptotically. In particular, our result confirms a conjecture of Mozafari-Nia and Iradmusa in the case m=n=3 in a strong sense.}, author = {Anastos, Michael and Boyadzhiyska, Simona and Rathke, Silas and Rué, Juanjo}, issn = {0166-218X}, journal = {Discrete Applied Mathematics}, pages = {506--511}, publisher = {Elsevier}, title = {{On the chromatic number of powers of subdivisions of graphs}}, doi = {10.1016/j.dam.2024.10.002}, volume = {360}, year = {2025}, } @article{18753, abstract = {We continue a line of research which studies which hereditary families of digraphs have bounded dichromatic number. For a class of digraphs C, a hero in C is any digraph H such that H -free digraphs in C have bounded dichromatic number. We show that if F is an oriented star of degree at least five, the only heroes for the class of F -free digraphs are transitive tournaments. For oriented stars F of degree exactly four, we show the only heroes in F -free digraphs are transitive tournaments, or possibly special joins of transitive tournaments. Aboulker et al. characterized the set of heroes of {H,K1+P2→} -free digraphs almost completely, and we show the same characterization for the class of {H,rK1+P3→} -free digraphs. Lastly, we show that if we forbid two "valid" orientations of brooms, then every transitive tournament is a hero for this class of digraphs.}, author = {Carbonero, Alvaro and Koerts, Hidde and Moore, Benjamin and Spirkl, Sophie}, issn = {0195-6698}, journal = {European Journal of Combinatorics}, publisher = {Elsevier}, title = {{On heroes in digraphs with forbidden induced forests}}, doi = {10.1016/j.ejc.2024.104104}, volume = {125}, year = {2025}, } @article{19002, abstract = {A k-subcolouring of a graph G is a function f : V (G) → {0,...,k − 1} such that the set of vertices coloured i induce a disjoint union of cliques. The subchromatic number, χsub(G), is the minimum k such that G admits a k-subcolouring. Nešetril, ˇ Ossona de Mendez, Pilipczuk, and Zhu (2020), recently raised the problem of finding tight upper bounds for χsub(G2) when G is planar. We show that χsub(G2) ≤ 43 when G is planar, improving their bound of 135. We give even better bounds when the planar graph G has larger girth. Moreover, we show that χsub(G3) ≤ 95, improving the previous bound of 364. For these we adapt some recent techniques of Almulhim and Kierstead (2022), while also extending the decompositions of triangulated planar graphs of Van den Heuvel, Ossona de Mendez, Quiroz, Rabinovich and Siebertz (2017), to planar graphs of arbitrary girth. Note that these decompositions are the precursors of the graph product structure theorem of planar graphs. We give improved bounds for χsub(Gp) for all p ≥ 2, whenever G has bounded treewidth, bounded simple treewidth, bounded genus, or excludes a clique or biclique as a minor. For this we introduce a family of parameters which form a gradation between the strong and the weak colouring numbers. We give upper bounds for these parameters for graphs coming from such classes. Finally, we give a 2-approximation algorithm for the subchromatic number of graphs having a layering in which each layer has bounded cliquewidth and this layering is computable in polynomial time (like the class of all dth powers of planar graphs, for fixed d). This algorithm works even if the power p and the graph G is unknown.}, author = {Cortés, Pedro P. and Kumar, Pankaj and Moore, Benjamin and Ossona de Mendez, Patrice and Quiroz, Daniel A.}, issn = {0012-365X}, journal = {Discrete Mathematics}, number = {4}, publisher = {Elsevier}, title = {{Subchromatic numbers of powers of graphs with excluded minors}}, doi = {10.1016/j.disc.2024.114377}, volume = {348}, year = {2025}, } @article{19017, abstract = {Let f(r)(n;s,k) denote the maximum number of edges in an n-vertex r-uniform hypergraph containing no subgraph with k edges and at most s vertices. Brown, Erdős and Sós [New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan 1971), pp. 53--63, Academic Press 1973] conjectured that the limit limn→∞n−2f(3)(n;k+2,k) exists for all k. The value of the limit was previously determined for k=2 in the original paper of Brown, Erdős and Sós, for k=3 by Glock [Bull. Lond. Math. Soc. 51 (2019) 230--236] and for k=4 by Glock, Joos, Kim, Kühn, Lichev and Pikhurko [arXiv:2209.14177, accepted by Proc. Amer. Math. Soc.] while Delcourt and Postle [arXiv:2210.01105, accepted by Proc. Amer. Math. Soc.] proved the conjecture (without determining the limiting value). In this paper, we determine the value of the limit in the Brown-Erdős-Sós Problem for k∈{5,6,7}. More generally, we obtain the value of limn→∞n−2f(r)(n;rk−2k+2,k) for all r≥3 and k∈{5,6,7}. In addition, by combining these new values with recent results of Bennett, Cushman and Dudek [arXiv:2309.00182] we obtain new asymptotic values for several generalised Ramsey numbers.}, author = {Glock, Stefan and Kim, Jaehoon and Lichev, Lyuben and Pikhurko, Oleg and Sun, Shumin}, issn = {1496-4279}, journal = {Canadian Journal of Mathematics}, pages = {1--43}, publisher = {Cambridge University Press}, title = {{On the (k + 2, k)-problem of Brown, Erdős, and Sós for k = 5,6,7}}, doi = {10.4153/s0008414x25000021}, year = {2025}, } @article{19018, abstract = {The online semi-random graph process is a one-player game which starts with the empty graph on n vertices. At every round, a player (called Builder) is presented with a vertex v chosen uniformly at random and independently from previous rounds, and constructs an edge of their choice that is incident to v. Inspired by recent advances on the semi-random graph process, we define a family of generalized online semi-random models. We analyse a particular instance that shares similar features with the original semi-random graph process and determine the hitting times of the classical graph properties minimum degree k,k-connectivity, containment of a perfect matching, a Hamiltonian cycle and an H-factor for a fixed graph H possessing an additional tree-like property. Along the way, we derive a few consequences of the famous Aldous-Broder algorithm that may be of independent interest.}, author = {Burova, Sofiya and Lichev, Lyuben}, issn = {0195-6698}, journal = {European Journal of Combinatorics}, publisher = {Elsevier}, title = {{The semi-random tree process}}, doi = {10.1016/j.ejc.2025.104120}, volume = {126}, year = {2025}, } @article{19418, abstract = {The size-Ramsey number r^(H) of a graph H is the smallest number of edges a (host) graph G can have, such that for any red/blue colouring of G, there is a monochromatic copy of H in G. Recently, Conlon, Nenadov and Trujić showed that if H is a graph on n vertices and maximum degree three, then r^(H)=O(n8/5), improving upon the upper bound of n5/3+o(1) by Kohayakawa, Rödl, Schacht and Szemerédi. In this paper we show that r^(H)≤n3/2+o(1). While the previously used host graphs were vanilla binomial random graphs, we prove our result using a novel host graph construction. Our bound hits a natural barrier of the existing methods.}, author = {Draganić, Nemanja and Petrova, Kalina H}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {3}, publisher = {Wiley}, title = {{Size‐Ramsey numbers of graphs with maximum degree three}}, doi = {10.1112/jlms.70116}, volume = {111}, year = {2025}, } @article{19433, abstract = {An ordered r-matching is an r-uniform hypergraph matching equipped with an ordering on its vertices. These objects can be viewed as natural generalisations of r-dimensional orders. The theory of ordered 2-matchings is well developed and has connections and applications to extremal and enumerative combinatorics, probability and geometry. On the other hand, in the case r≥3 much less is known, largely due to a lack of powerful bijective tools. Recently, Dudek, Grytczuk and Ruciński made some first steps towards a general theory of ordered r-matchings, and in this paper we substantially improve several of their results and introduce some new directions of study. Many intriguing open questions remain.}, author = {Anastos, Michael and Jin, Zhihan and Kwan, Matthew Alan and Sudakov, Benny}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Extremal, enumerative and probabilistic results on ordered hypergraph matchings}}, doi = {10.1017/fms.2024.144}, volume = {13}, year = {2025}, } @article{19440, abstract = {Let μ(G) denote the minimum number of edges whose addition to G results in a Hamiltonian graph, and let μ^(G) denote the minimum number of edges whose addition to G results in a pancyclic graph. We study the distributions of μ(G),μ^(G) in the context of binomial random graphs. Letting d=d(n):=n⋅p, we prove that there exists a function f:R+→[0,1] of order f(d)=12de−d+e−d+O(d6e−3d) such that, if G∼G(n,p) with 20≤d(n)≤0.4logn, then with high probability μ(G)=(1+o(1))⋅f(d)⋅n. Let ni(G) denote the number of degree i vertices in G. A trivial lower bound on μ(G) is given by the expression n0(G)+⌈12n1(G)⌉. In the denser regime of random graphs, we show that if np−13logn−2loglogn→∞ and G∼G(n,p) then, with high probability, μ(G)=n0(G)+⌈12n1(G)⌉. For completion to pancyclicity, we show that if G∼G(n,p) and np≥20 then, with high probability, μ^(G)=μ(G). Finally, we present a polynomial time algorithm such that, if G∼G(n,p) and np≥20, then, with high probability, the algorithm returns a set of edges of size μ(G) whose addition to G results in a pancyclic (and therefore also Hamiltonian) graph.}, author = {Alon, Yahav and Anastos, Michael}, issn = {1098-2418}, journal = {Random Structures and Algorithms}, number = {2}, publisher = {Wiley}, title = {{The completion numbers of hamiltonicity and pancyclicity in random graphs}}, doi = {10.1002/rsa.21286}, volume = {66}, year = {2025}, } @article{19503, abstract = {A tantalizing open problem, posed independently by Stiebitz in 1995 and by Alon in 1996 and again in 2006, asks whether for every pair of integers s,t≥1 there exists a finite number F(s,t) such that the vertex set of every digraph of minimum out-degree at least F(s,t) can be partitioned into non-empty parts A and B such that the subdigraphs induced on A and B have minimum out-degree at least s and t , respectively. In this short note, we prove that if F(2,2) exists, then all the numbers F(s,t) with s,t≥1 exist and satisfy F(s,t)=Θ(s+t) . In consequence, the problem of Alon and Stiebitz reduces to the case s=t=2 . Moreover, the numbers F(s,t) with s,t≥2 either all exist and grow linearly, or all of them do not exist.}, author = {Christoph, Micha and Petrova, Kalina H and Steiner, Raphael}, issn = {1469-2163}, journal = {Combinatorics Probability and Computing}, number = {4}, pages = {559--564}, publisher = {Cambridge University Press}, title = {{A note on digraph splitting}}, doi = {10.1017/S0963548325000045}, volume = {34}, year = {2025}, } @article{19508, abstract = {We consider random two-player zero-sum dynamic games with perfect information on a class of infinite directed graphs. Starting from a fixed vertex, the players take turns to move a token along the edges of the graph. Every vertex is assigned a payoff known in advance by both players. Every time the token visits a vertex, Player 2 pays Player 1 the corresponding payoff. We consider a distribution over such games by assigning i.i.d. payoffs to the vertices. On the one hand, for acyclic directed graphs of bounded degree and sub-exponential expansion, we show that, when the duration of the game tends to infinity, the value converges almost surely to a constant at an exponential rate dominated in terms of the expansion. On the other hand, for the infinite d-ary tree (that does not fall into the previous class of graphs), we show convergence at a double-exponential rate.}, author = {Attia, Luc and Lichev, Lyuben and Mitsche, Dieter and Saona Urmeneta, Raimundo J and Ziliotto, Bruno}, issn = {2153-0793}, journal = {Dynamic Games and Applications}, pages = {1517--1535}, publisher = {Springer Nature}, title = {{Random zero-sum dynamic games on infinite directed graphs}}, doi = {10.1007/s13235-025-00636-4}, volume = {15}, year = {2025}, } @article{19554, abstract = {In 1981, Karp and Sipser proved a law of large numbers for the matching number of a sparse Erdős–Rényi random graph, in an influential paper pioneering the so-called differential equation method for analysis of random graph processes. Strengthening this classical result, and answering a question of Aronson, Frieze and Pittel, we prove a central limit theorem in the same setting: the fluctuations in the matching number of a sparse random graph are asymptotically Gaussian. Our new contribution is to prove this central limit theorem in the subcritical and critical regimes, according to a celebrated algorithmic phase transition first observed by Karp and Sipser. Indeed, in the supercritical regime, a central limit theorem has recently been proved in the PhD thesis of Kreačić, using a stochastic generalisation of the differential equation method (comparing the so-called Karp–Sipser process to a system of stochastic differential equations). Our proof builds on these methods, and introduces new techniques to handle certain degeneracies present in the subcritical and critical cases. Curiously, our new techniques lead to a non-constructive result: we are able to characterise the fluctuations of the matching number around its mean, despite these fluctuations being much smaller than the error terms in our best estimates of the mean. We also prove a central limit theorem for the rank of the adjacency matrix of a sparse random graph.}, author = {Glasgow, Margalit and Kwan, Matthew Alan and Sah, Ashwin and Sawhney, Mehtaab}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {4}, publisher = {Wiley}, title = {{A central limit theorem for the matching number of a sparse random graph}}, doi = {10.1112/jlms.70101}, volume = {111}, year = {2025}, } @article{19603, abstract = {MaxCut is a classical NP-complete problem and a crucial building block in many combinatorial algorithms. The famousEdwards-Erdös bound states that any connected graph on n vertices with m edges contains a cut of size at least m/2 + n−1/4 . Crowston, Jones and Mnich [Algorithmica, 2015] showed that the MaxCut problem on simple connected graphs admits an FPT algorithm, where the parameter k is the difference between the desired cut size c and the lower bound given by the Edwards-Erdös bound. This was later improved by Etscheid and Mnich [Algorithmica, 2017] to run in parameterized linear time, i.e., f (k) · O(m). We improve upon this result in two ways: Firstly, we extend the algorithm to work also for multigraphs (alternatively, graphs with positive integer weights). Secondly, we change the parameter; instead of the difference to the Edwards-Erdös bound, we use the difference to the Poljak-Turzík bound. The Poljak-Turzík bound states that any weighted graph G has a cut of weight at least w(G)/2 + wMSF (G)/4 , where w(G) denotes the total weight of G, and wMSF (G) denotes the weight of its minimum spanning forest. In connected simple graphs the two bounds are equivalent, but for multigraphs the Poljak-Turzík bound can be larger and thus yield a smaller parameter k. Our algorithm also runs in parameterized linear time, i.e., f (k) · O(m + n).}, author = {Lill, Jonas and Petrova, Kalina H and Weber, Simon}, issn = {1432-0541}, journal = {Algorithmica}, pages = {983--1007}, publisher = {Springer Nature}, title = {{Linear-time MaxCut in multigraphs parameterized above the Poljak-Turzík bound}}, doi = {10.1007/s00453-025-01306-y}, volume = {87}, year = {2025}, } @article{19798, abstract = {In an n×n array filled with symbols, a transversal is a collection of entries with distinct rows, columns and symbols. In this note we show that if no symbol appears more than βn times, the array contains a transversal of size (1−β/4−o(1))n . In particular, if the array is filled with n symbols, each appearing n times (an equi- n square), we get transversals of size (3/4−o(1))n. Moreover, our proof gives a deterministic algorithm with polynomial running time, that finds these transversals.}, author = {Anastos, Michael and Morris, Patrick}, issn = {1520-6610}, journal = {Journal of Combinatorial Designs}, number = {9}, pages = {338--342}, publisher = {Wiley}, title = {{A note on finding large transversals efficiently}}, doi = {10.1002/jcd.21990}, volume = {33}, year = {2025}, }