In every 3-connected graph, some longest cycle has a chord. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
The covering number of a family is the size of the smallest set that intersects all sets from the family. In 1978 Frankl determined for n >= n(0)(k) the largest intersecting family of kelement subsets of [n] with covering number 3. In this paper, we essentially settle this problem, showing that the same family is extremal for any k >= 100 and n > 2k. (c) 2025 Published by Elsevier Inc.
We prove that for any two positive integers k1 and k2, if G is a graph, S1, T1, S2, T2 are vertex-subsets of G, and G is edge-minimal with respect to the condition that for i = 1, 2 there are ki disjoint paths of G between Si and Ti, then G contains at most 12k1k2 vertices of degree at least three. This bound is optimal up to a constant factor, as a k1 x k2-grid G shows. The degree-3 treewidth sparsifier theorem, proved by Chekuri and Chuzhoy (2015), states that for any graph G of treewidth at least k, there is a subcubic subgraph of G that has treewidth Omega(k/ poly log k) and contains O(k4) vertices of degree three. Our result, together with their proof techniques, reduces the bound O(k4) in this theorem to O(k2), solving their conjecture. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
For graphs F and H, let i(F) denote the inducibility of F and let iH(F) denote the inducibility of F over H-free graphs. We prove that for almost all graphs F on a given number of vertices, iKk(F) attains infinitely many values as k varies. For complete partite graphs F (and, more generally, for symmetrizable families of graphs F), we prove that iH(F)=iKk(F) where k=χ(H), and is attained by a complete ℓ-partite graphon WF,k, where ℓ<k.We determine the part sizes of WF,k for all k, whence determine i(F), whenever F is the Turán graph on s vertices and r parts, for all s≤3r+1, which was recently proved by Liu, Mubayi, and Reiher for s=r+1. As a corollary, this determines the inducibility of all Turán graphs on at most 14 vertices. Furthermore, since inducibility is invariant under complement, this determines the inducibility of all matchings and, more generally, all graphs with maximum degree 1, of any size. Similarly, this determines the inducibility of all triangle factors, of any size.For complete partite graphs F with at most one singleton part, we prove that iKk(F) only attains finitely many values as k varies; in particular, there exists t=t(F) such that i(F) is attained by some complete t-partite graphon. This is best possible as it was shown by Liu, Pikhurko, Sharifzadeh, and Staden that this is not necessarily true if there are two singleton parts.Finally, for every r, we give a nontrivial sufficient condition for a complete r-partite graph F to have the property that i(F) is attained by a complete partite graphon all whose part sizes are distinct.
Let k > 2 be an integer. A digraph D is k-linked if for every set of 2k distinct vertices x1, ..., xk, y1,..., yk in D, there exist k pairwise vertex-disjoint paths P-1, ..., P-k such that each path Pi starts at xiand ends at yi for i is an element of [k]. In 2015, Pokrovskiy conjectured that there exists a function g(k) such that every 2k-connected tournament with minimum in-degree and minimum out-degree at least g(k) is k-linked in Pokrovskiy (2015) [16]. In this paper, we disprove Pokrovskiy's conjecture by constructing a family of 2k-connected tournaments of order n >= 14k(2 )with arbitrarily large minimum semi-degree (depending on n) that are not klinked. The counterexamples, with sufficiently large order n, also provide a negative answer to the question posed by Girao et al. (2021) [8]: whether or not 2k-connectivity is sufficient for k-linkage in every tournament with minimum out-degree at least some polynomial in k.
A k-graph H is called (p, mu)-dense if for all not necessarily distinct sets A(1), ... , A(k) subset of V(H) we have e(A(1), ... , A(k)) > p|A(1)||A(k)|- mu |V(H)|(k). This is believed to be the weakest form of quasirandomness in k-graphs and also known as linear quasirandomness. In this paper, we show that for & ell; < k satisfying (k-& ell;) k, (p, mu)-density plus a minimum (& ell; + 1)-degree of alpha n(k-& ell;-1) guarantees Hamilton & ell;-cycles, but requiring a minimum & ell;-degree of Omega(n(k-& ell;)) instead is not sufficient. This answers a question of Lenz-Mubayi-Mycroft and characterizes the triples (k, & ell;, d) when k-& ell; k such that degenerate choices of p and alpha force & ell;-Hamiltonicity. We actually prove a general result on & ell;-Hamiltonicity in quasirandom k-graphs, assuming a minimum vertex degree and essentially that every two & ell;-sets can be connected by a constant length & ell;-path. This reduces the & ell;-Hamiltonicity problem to the study of the connection property which also allows us to deduce a (k/2)-Hamiltonicity result in uniformly dense k-graphs (for even k >= 4). Our proof uses the lattice-based absorption method in the non-standard way and is the first one that embeds a nonlinear Hamilton cycle in linear quasirandom k-graphs. (c) 2025 Published by Elsevier Inc.
We prove that if G is a 2-connected graph with minimum degree at least k⩾4, then(1)G contains k cycles whose lengths form an arithmetic progression with common difference one or two, unless G≅Kk+1 or Kk,n−k;(2)G contains cycles of lengths ℓ modulo k for all even ℓ, unless G≅Kk+1 or Kk,n−k;(3)G contains cycles of lengths ℓ modulo k for all ℓ, unless G≅Kk+1 or G is bipartite. In addition, we show that if k is even and G is 2-connected with minimum degree at least k−1 and order at least k+2, then G contains cycles of lengths ℓ modulo k for all even ℓ. As a corollary, we determine the maximum number of edges in a graph without a cycle of length divisible by k for all odd k.
In 2002, Nikiforov proved that for an n-vertex graph C with clique number omega and edge number m, its spectral radius lambda(G) satisfies lambda(G) <= root 2(1-1/omega)m, which confirmed a conjecture implicitly suggested by Edwards and Elphick. In this paper, we prove a local version of spectral Tur & aacute;n inequality, showing that lambda(2)(G) <= 2 Sigma(e is an element of E(G)) c(e)-1/c(e) , where c(e) is the order of the e is an element of E(G) largest clique containing the edge e in G. We also characterize the extremal graphs. Furthermore, we prove that our theorem implies Nikiforov's theorem and provide an example in which the difference of Nikiforov's bound and ours is Omega(root m) for some cases. Our second result explores local properties of the Perron vector of graphs. We disprove a conjecture of Gregory, asserting that for a connected n-vertex graph G with chromatic number k >= 2 and an independent set S, we have (Sigma v is an element of S) x(v)(2) <= 1/ 2- k-2/2 root (k-2)(2) + 4(k-1)(n-k + 1), where x(v) is the component of the Perron vector of G with respect to the vertex v. A modified version of Gregory's conjecture is proposed. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We show that the k-colour Ramsey number of an odd cycle of length 2 & ell; + 1 is at most (4 & ell;)k & centerdot; kk/& ell;. This proves a conjecture of Fox and is the first improvement in the exponent that goes beyond an absolute constant factor since the work of Bondy and Erd & odblac;s from 1973. (c) 2026 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
alpha(Fqd,p) general position in a p-random subset of Fqd. We determine the order of magnitude of alpha(Fqd,p) up to a polylogarithmic factor by proving the balanced supersaturation conjecture of Balogh and Luo. Our result also resolves a conjecture implicitly posed by the first author, Liu, the second author and Zeng. In the course of our proof, we establish a lemma that demonstrates a "structure vs. randomness" phenomenon for point sets in finite-field linear spaces, which may be of independent interest. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Two families A and B are cross-intersecting if A boolean AND B not equal theta for any A is an element of A and B is an element of B. We call t families A(1), A(2), ... , A(t) pairwise cross-intersecting families if A(i) and A(j) are cross-intersecting for 1 <= i < j <= t. Additionally, if A(j) not equal theta for each j is an element of [t], then we say that A(1), A(2), ... , A(t) are non-empty pairwise cross-intersecting. Let A(1) subset of ((k1)[(n)]), A(2) subset of([(n)(k2)]),...,A(t) subset of ([(n)(kt)]) be non-k1 k2 kt empty pairwise cross-intersecting families with t >= 2, k(1) >= k(2) > center dot center dot center dot > k(t), and n >= k(1) +k(2), we determine the maximum of Sigma(t)(i=1) |A(i)| and characterize all extremal families. This answers a question of Shi, Frankl and Qian [Combinatorica 42 (2022)]. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Spectral graph theory studies how the eigenvalues of a graph relate to the structural properties of a graph. In this paper, we solve three open problems in spectral extremal graph theory which generalize the classical Tur & aacute;n-type supersaturation results. center dot We prove that every m-edge graph G with the spectral radius lambda(G) >root m contains at least 1/24 root m triangles 24 sharing a common edge. This result confirms a conjecture of Nikiforov, and Li and Peng. Moreover, the bound is optimal up to a constant factor. center dot For m-edge graph G with lambda(G) > root(1-1/r)2m, we show that it must contain Omega(r)(/m) copies of Kr +1 sharing r common vertices. This confirms a conjecture of Li, Liu and Feng and unifies a series of spectral extremal results on books and cliques. Moreover, we also show that such a graph G contains Omega(r)(m (r-1/2) ) copies of Kr +1. This extends a result of Ning and Zhai for counting triangles. center dot We prove that every m-edge graph G with lambda(G) > root m contains at least (1/8-o(1))m(2) copies of 4-cycles, and we provide two constructions showing that the constant 1/8 is the best possible. This result settles a problem raised by Ning and Zhai, and it gives the first asymptotics for counting degenerate bipartite graphs. The key to our proof is two structural results we obtain for graphs with large spectral radii on their maximum degree and on existence of large structured subgraphs, which we believe to be of independent interest. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We prove a coarse version of Halin's Grid Theorem: Every one-ended, locally finite graph that contains the disjoint union of infinitely many rays as an asymptotic minor also contains the half-grid as an asymptotic minor. More generally, we show that the same holds for arbitrary (not necessarily one-ended or locally finite) graphs under additional, necessary assumptions on the minor-models of the infinite rays. This resolves a conjecture of Georgakopoulos and Papasoglu. As an application, we show that every one-ended, quasi-transitive, locally finite graph contains the half-grid as an asymptotic minor and as a diverging minor. This in particular includes all locally finite Cayley graphs of one-ended finitely generated groups and solves a problem of Georgakopoulos and Papasoglu.
By unifying various earlier extensions of alternating sign matrices (ASMs), we introduce the notion of prefix-bounded matrices (PBMs). It is shown that the convex hull of these matrices forms the intersection of two special generalized polymatroids. This implies — in a more general form — that the linear inequality system given by Behrend and Knight [3] and by Striker [37], [38] for describing the polytope of alternating sign matrices is totally dual integral (TDI), confirming a recent conjecture of Edmonds [13], [14]. By relying on the polymatroidal approach, we derive a characterization for the existence of prefix-bounded matrices meeting lower and upper bounds on their entries.Furthermore, we point out that the constraint matrix of the linear system describing the convex hull of PBMs, in particular ASMs, is a network matrix. This implies that (a) standard network-flow techniques can be used to algorithmically handle optimization problems and structural results on PBMs obtained via g-polymatroids, (b) the linear system is actually box-TDI, and (c) the convex hull of PBMs admits a sharpened form of the integer Carathéodory property, in particular, the integer decomposition property. The latter feature makes it possible to confirm an extended form of an elegant conjecture of Brualdi and Dahl [7] on the decomposability of a so-called k-regular alternating sign matrix as the sum of k pattern-disjoint ASMs.
We address a problem posed by Erdős and Hajnal in 1991, proving that for all n ≥ 600, every (2n+1)-vertex graph with at least n^2 + n + 1 edges contains two vertices of equal degree connected by a path of length three. The complete bipartite graph K_n,n+1 demonstrates that this edge bound is sharp. We further establish an analogous result for graphs with even order and investigate several related extremal problems.
In this paper, we provide the first known infinite family of simple graphs, each of which is the skeleton of a chiral map, a skeleton of a reflexible map on an orientable surface, as well as a skeleton of a reflexible map on a non-orientable surface. This family consists of all lexicographic products Cn[mK1], where m >= 3, n = sm, with s an integer not divisible by 4. This answers a question posed by Wilson in 2002. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
We prove that, for all k >= 3, and any integers Delta, n with n >= Delta, there exists a k-uniform hypergraph on n vertices with maximum degree at most Delta whose 4-color Ramsey number is at least tw(k)(c(k)Delta) & centerdot; n, for some constant ck > 0, where tw(k) denotes the tower function. For k >= 4, this is tight up to the constant c(k) and for k = 3 it is known to be tight up to a factor of log Delta on top of the tower. Our bound extends a well-known result of Graham, R & ouml;dl and Ruci & nacute;ski for graphs and answers a question of Conlon, Fox and Sudakov from 2009 for four colors. (c) 2026 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
