We introduce the notion of delineation. A graph class C is said delineated by twin-width (or simply, delineated) if for every hereditary closure D of a subclass of C, it holds that D has bounded twin-width if and only if D is monadically dependent. An effective strengthening of delineation for a class C implies that tractable FO model checking on C is perfectly understood: On hereditary closures D of subclasses of C, FO model checking is fixed-parameter tractable (FPT) exactly when D has bounded twin-width. Ordered graphs [BGOdMSTT, STOC ’22], permutation graphs [BKTW, JACM ’22] and tournaments [GT, European Journal of Combinatorics ’25] are effectively delineated, while subcubic graphs are not. On the one hand, we prove that interval graphs, and even, directed path graphs with boundedly many roots are delineated. On the other hand, we observe or show that segment graphs, directed path graphs (with arbitrarily many roots), and visibility graphs of simple polygons are not delineated.
This paper establishes a linear bound in n for the difference between the average non-orientable genus and maximum nonorientable genus of a random graph on n vertices. This result demonstrates that the average non-orientable genus of a random graph asymptotically equals its maximum genus, thus resolving a conjecture by Archdeacon. To achieve this result, we utilize pre-signed graphs-a combinatorial framework that generalizes permutation-partition pairs to unify signed graphs and their embeddings. (c) 2026 Published by Elsevier Ltd.
The Hoffman problem of characterizing graphs whose second largest eigenvalue does not exceed 1 has attracted a great deal of attention over the past four decades, although a complete characterization remains open for general graphs. In this paper, we focus particularly on K4-minor free graphs with the second largest eigenvalue not exceeding 1. Our result includes all connected outerplanar graphs and generalized theta-graphs with the same spectral property, and therefore generalizes the result of Li and Sun (2023) and the result of Gao and Huang (2008). (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The purpose of this note is to extend previous known results about maximal chains of copies of countable ultrahomogeneous relational structures. In particular, we completely describe order types of maximal chains in posets of the form P(P) _ {A subset of P: A similar to_ P}, where P is a countable ultrahomogeneous digraph belonging to Cherlin's fourth class, freely generated countable ultrahomogeneous directed graphs. Analogous characterizations were known for countable ultrahomogeneous posets and graphs. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We prove equidistribution of two pairs of statistics on boxed plane partitions: (volume, trace) and (corner-hook volume, number of corners). The proof relies on different 3d visualizations of the corresponding non-intersecting path systems. In particular, we obtain a new visual proof for a volume generating function of plane partitions. We also introduce a new statistic called the cohook area on ordinary partitions, and prove that it is equidistributed with the area of partitions.
Let H be obtained from a cyclically 4-edge-connected cubic planar graph Y other than K4 by deleting two adjacent vertices. We provide a short proof that if H has circumference at least k for some even integer k >= 4, then H contains a cycle of length between k and 3k/2. As a consequence, we show that the line graph G of Y contains a cycle of length l avoiding any prescribed vertex of G, for every l is an element of {3} boolean OR {5, ... , |V(G)| - 1}. The proofs integrate Euler's formula and the Three Edge Lemma, established by Thomas and Yu, and independently by Sanders, in a novel way. This work was partially motivated by conjectures of Bondy and Malkevitch. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Let fW(n) be the number of different factors of length n appearing in a word W. In a classical result by Morse and Hedlund, a characterization is given for infinite words satisfying the condition f(W)(n) <= n for some n E N. In this paper, we describe the form of finite words that satisfy the condition fW(n) <= n. We study relations between power avoidance and subword complexity of a finite word. We apply our combinatorial results to study the interrelations between various numerical invariants of finite-dimensional associative algebras. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Let p(t)(star)(n) (resp. sc(t)(star) (n) and dd(t)(star)(n)) be the total number of t-hooks among the ordinary (resp. self-conjugate and doubled distinct) partitions of n. We combinatorially derive the generating functions for p(t)(star)(n), sc(t)(star)(n), and dd(t)(star)(n). By constructing combinatorial injections using these generating functions, we investigate the monotonicity of these three counting functions according to n. In particular, we show that p(t)(star)(n) >= p(t)(star)+1(n) except for n = t + 1, sc(2k)(star)-1(n) >= sc(2k)(star)(n) for n > 84k, k >= 1, dd(1)(star)(n) >= dd(2)(star)(n) except for n = 6, 10, and dd(2k)(star)(n) >= dd(2k+1)(star)(n) except for (k, n) = (1, 12), (2, 10), (3, 14). (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We construct connected 2-arc-transitive covers of complete graphs with non-abelian characteristically simple transformation groups. This solves the existence problem for non-solvable 2-arc-transitive covers of complete graphs.
The matching conjecture of Erd & odblac;s concerns the maximum number of edges in an r-uniform graph with given matching number. It is natural to determine the maximum number of other substructures. Let Sr-r-1,Sr-2 denote the r-uniform graph consisting of 2 edges intersecting at r-1 vertices. For an r-uniform graph 9C and a subgraph S C 9C, let N(S, 9C) denote the number of isomorphic copies of S in 9C. We show that if s, r > 2, n > 2sr3 and 9C C ([n]) has matching number less than r s, then N(Sr-r-1,Sr-2, 9C) <_ N(Sr-r-1,Sr-2, ([n])-([s,n] equality holds if and only if 9C = ([n] )). Moreover, the r )-([n]\S r ) for some S E ([n]).Counting the maximum number of copies of Sr rr s-1 r-1,2 in runiform graphs with given matching number is also related to the hypergraph Tur & aacute;n number of a matching in 2-norm pound called by Balogh et al. (2022). Combining with the result of Frankl (2013), our result implies a result by Brooks and Linz (0000) concerning the Erd & odblac;s' matching conjecture in e(2)-norm (in a stronger sense, the condition on n in Brooks and Linz (0000) is sufficiently large), and the result in (2)-norm pound Brooks and Linz (0000) does not imply our result. For the quantitative part of the main result, we may assume that a maximum hypergraph is shifted in the proof. Comparing to count the maximum number of edges, we need to overcome two difficulties in counting the maximum number of copies of Sr-r-1,Sr-2. Firstly, we show that shifting an r-uniform graph will not decrease the number of copies of Sr-r-1,Sr-2 by establishing an injection. Secondly, we give nontrivial methods to count the number of copies of Sr-r-1,Sr-2. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The equivalence of group connectivity for non-homogeneous groups with a same order has been concerned since Jaeger, Linial, Payan and Tarsi introduced this concept in [J. Combin. Theory Ser. B, 56 (1992) 165-182]. Hu & scaron;ek, Moheln & iacute;kov & aacute; and & Scaron;& aacute;mal in [J. Graph Theory, 93 (2020) 317-327] showed that Z4-connectivity and Z22-connectivity are not equivalent by finding counterexamples with a computer-assisted proof, and they asked whether one can find a proof that does not use computers. Langhede and Thomassen [European J. Combin., (2023) 103816] provide a computer-free proof to show that there exist 3-edge-connected, Z22-connected, and non-Z4-connected graphs. In this paper, we construct 3-edge-connected graphs which are Z4-connected but not Z22-connected in which we prove those properties without any involvement of computers. These two results together answer the question proposed by Hu & scaron;ek et al. about computer-free proofs on the non-equivalence of Z4-connectivity and Z22-connectivity. In addition, by using both theoretical reductions and computer searching we find the smallest graph whose Z4-connectivity varies from Z22-connectivity. This smallest graph (in terms of order and size) is unique, which has 10 vertices and 14 edges. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The basis number of a graph G is the smallest integer k such that G admits a basis B for its cycle space, where each edge of G belongs to at most k members of B. In this note, we show that every non-planar graph that can be embedded on a surface with Euler characteristic 0 has a basis number of exactly 3, proving a conjecture of Schmeichel from 1981. Additionally, we show that any graph embeddable on a surface Sigma (whether orientable or non-orientable) of genus g has a basis number of O(log(g)2). (c) 2026 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
The codegree squared sum co2(F) of a family (hypergraph) F⊆([n]k) is defined to be the sum of codegrees squared d(E)2 over all E∈([n]k−1), where d(E)=|{F∈F:E⊆F}|. Given a family of k-uniform hypergraphs ℋ, Balogh, Clemen and Lidický recently introduced the problem to determine the maximum codegree squared sum co2(F) over all ℋ-free F. In the present paper, we consider the families which have as forbidden configurations all pairs of sets with intersection sizes less than t, that is, the well-known t-intersecting families. We prove the following Erdős–Ko–Rado Theorem in ℓ2-norm, which confirms a conjecture of Brooks and Linz.Let t,k,n be positive integers such that t≤k≤n. If a family F⊆([n]k) is t-intersecting, then for n≥(t+1)(k−t+1), we have co2(F)≤(n−tk−t)(t+(n−k+1)(k−t)),equality holds if and only if F={F∈([n]k):T⊆F} for some t-subset T of [n].In addition, we prove a Frankl–Hilton–Milner Theorem in ℓ2-norm for t≥2, and a generalized Turán result, i.e., we determine the maximum number of copies of tight path of length 2 in t-intersecting families.
The classical Erd & odblac;s-Ko-Rado theorem on the size of an intersecting family of k-subsets of the set [n] = {1, 2, ..., n} is one of the fundamental intersection theorems for set systems which has had great impact on combinatorics. In 1995, Snevily proposed the conjecture that the upper bound for the size of an L-intersecting family of subsets of [n] is (n ) under the condition smax{& ell;(i)} < min{k(j)}, where L = {& ell;(1), ..., & ell;(s)} with 0 < & ell;(1) < & centerdot; & centerdot; & centerdot; < & ell;(s) and kj are subset sizes in the family. This conjecture has been shown to be true when n is sufficiently large or when L = {0, 1, ..., s-1}. In this paper, we provide a vector space generalization of Snevily's conjecture and show that this generalized conjecture holds when L = {0, 1, ..., s-1}. We then provide a strengthening of this generalized conjecture which unifies several well-known theorems, including the Erd & odblac;s-Ko-Rado theorem for vector spaces, and confirm the strengthened conjecture for sufficiently large n. Our results unify or strengthen most of the existing related theorems. Moreover, we characterize the extremal L-intersecting families of subspaces of an n-dimensional vector space over a finite field. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The equivalence of group connectivity for non-homogeneous groups with a same order has been concerned since Jaeger, Linial, Payan and Tarsi introduced this concept in [J. Combin. Theory Ser. B, 56 (1992) 165-182]. Hušek, Mohelníková and Šámal in [J. Graph Theory, 93 (2020) 317-327] showed that Z4-connectivity and Z22-connectivity are not equivalent by finding counterexamples with a computer-assisted proof, and they asked whether one can find a proof that does not use computers. Langhede and Thomassen [European J. Combin., (2023) 103816] provide a computer-free proof to show that there exist 3-edge-connected, Z22-connected, and non-Z4-connected graphs. In this paper, we construct 3-edge-connected graphs which are Z4-connected but not Z22-connected in which we prove those properties without any involvement of computers. These two results together answer the question proposed by Hušek et al. about computer-free proofs on the non-equivalence of Z4-connectivity and Z22-connectivity. In addition, by using both theoretical reductions and computer searching we find the smallest graph whose Z4-connectivity varies from Z22-connectivity. This smallest graph (in terms of order and size) is unique, which has 10 vertices and 14 edges.
A hyperfield H is stringent if a boxed plus b is a singleton unless a = -b, for all a, b is an element of H. By a construction of Marc Krasner, each valued field gives rise to a stringent hyperfield. We show that if H is a stringent skew hyperfield, then weak matroids over H are strong matroids over H. Also, we present vector axioms for matroids over stringent skew hyper-fields which generalize the vector axioms for oriented matroids and valuated matroids. (c) 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
