We present and prove closed form expressions for some families of binomial determinants with signed Kronecker deltas that are located along an arbitrary diagonal in the corresponding matrix. They count cyclically symmetric rhombus tilings of hexagonal regions with triangular holes. We extend a previous systematic study of these families, where the locations of the Kronecker deltas depended on an additional

The Turán number of hypergraphs has been studied extensively. Here we deal with a recent direction, the linear Turán number, and restrict ourselves to linear triple systems, a collection of triples on a set of points in which any two triples intersect in at most one point. For a fixed linear triple system F, the linear Turán number exL(n,F) is the maximum number of triples in a linear triple system

In this paper, we attach several new invariants to connected strongly regular graphs (excepting conference graphs on non-square number of vertices): one invariant called the discriminant, and a p-adic invariant corresponding to each prime number p. We prove parametric restrictions on quasi-symmetric 2-designs with a given connected block graph G and a given defect (absolute difference of the two intersection

Since 1997 a considerable effort has been spent to study the mixing time of switch Markov chains on the realizations of graphic degree sequences of simple graphs. Several results were proved on rapidly mixing Markov chains on unconstrained, bipartite, and directed sequences, using different mechanisms. The aim of this paper is to unify these approaches. We will illustrate the strength of the unified

Permutation statistics wm¯ and rlm are both arising from permutation tableaux. wm¯ was introduced by Chen and Zhou, which was proved equally distributed with the number of unrestricted rows of a permutation tableau. While rlm is shown by Nadeau equally distributed with the number of 1’s in the first row of a permutation tableau. In this paper, we investigate the joint distribution of wm¯ and rlm. Statistic

Penttila and Williford constructed a 4-class association scheme from a generalized quadrangle with a doubly subtended subquadrangle. We show that an association scheme with appropriate parameters and satisfying some assumption about maximal cliques must be the Penttila–Williford scheme.

We consider quantitative versions of Helly-type questions, that is, instead of finding a point in the intersection, we bound the volume of the intersection. Our first main result is a quantitative version of the Fractional Helly Theorem of Katchalski and Liu, the second one is a quantitative version of the (p,q)-Theorem of Alon and Kleitman.

We study the problems of bounding the number weak and strong independent sets in r-uniform, d-regular, n-vertex linear hypergraphs with no cross-edges. In the case of weak independent sets, we provide an upper bound that is tight up to the first order term for all (fixed) r≥3, with d and n going to infinity. In the case of strong independent sets, for r=3, we provide an upper bound that is tight up

The semi-random graph process is a single player game in which the player is initially presented an empty graph on n vertices. In each round, a vertex u is presented to the player independently and uniformly at random. The player then adaptively selects a vertex v, and adds the edge uv to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this

We show that A-paths of length 0 modulo 4 have the Erdős–Pósa property. Modulus m=4 is the only composite number for which A-paths of length 0 modulo m have the property.

The odd wheel W2k+1 is the graph formed by joining a vertex to a cycle of length 2k. In this paper, we investigate the largest value of the spectral radius of the adjacency matrix of an n-vertex graph that does not contain W2k+1. We determine the structure of the spectral extremal graphs for all k≥2,k⁄∈{4,5}. When k=2, we show that these spectral extremal graphs are among the Turán-extremal graphs

The distinguishing index D′(Γ) of a graph Γ is the least number k such that Γ has an edge-coloring with k colors preserved only by the trivial automorphism. In this paper we prove that if the automorphism group of a finite graph Γ is simple, then its distinguishing index D′(Γ)=2.

Motivated by a polynomial identity of certain iterated integrals, first observed in Colmenarejo et al. (2020) in the setting of lattice paths, we prove an intriguing combinatorial identity in the shuffle algebra. It has a close connection to de Bruijn’s formula when interpreted in the framework of signatures of paths.

We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are knotoid isotopy invariant. Secondly, we define finite type invariants directly on knotoids, by extending knotoid invariants to singular knotoid invariants via the

Words are sequences of letters over a finite alphabet. We study two intimately related topics for this object: quasi-randomness and limit theory. With respect to the first topic we investigate the notion of uniform distribution of letters over intervals, and in the spirit of the famous Chung–Graham–Wilson theorem for graphs we provide a list of word properties which are equivalent to uniformity. In

We study ν-Schröder paths, which are Schröder paths which stay weakly above a given lattice path ν. Some classical bijective and enumerative results are extended to the ν-setting, including the relationship between small and large Schröder paths. We introduce two posets of ν-Schröder objects, namely ν-Schröder paths and trees, and show that they are isomorphic to the face poset of the ν-associahedron

Probabilistic zero forcing is a coloring game played on a graph where the goal is to color every vertex blue starting with an initial blue vertex set. As long as the graph G is connected, if at least 1 vertex is blue then eventually all of the vertices will be colored blue. The most studied parameter in probabilistic zero forcing is the expected propagation time ept(G). We significantly improve on

For a fixed set of positive integers R, we say H is an R-uniform hypergraph, or R-graph, if the cardinality of each edge belongs to R. For a graph G=(V,E), a hypergraph H is called a Berge-G, if there is an injection i:V(G)→V(H) and a bijection f:E(G)→E(H) such that for all e=uv∈E(G), we have {i(u),i(v)}⊆f(e). We define the cover Turán number of a graph G, denoted as exˆR(n,BG), as the maximum number

Let p be an odd prime. From a simple undirected graph G, through the classical procedures of Baer (1938), Tutte (1947) and Lovász (1989), there is a p-group PG of class 2 and exponent p that is naturally associated with G. Our first result is to show that this construction of groups from graphs respects isomorphism types. That is, given two graphs G and H, G and H are isomorphic as graphs if and only

We correct an error in proof of the author’s Theorem [1, Thm. 4.1].

A digraph is k-linked if for any two disjoint sets of vertices {x1,…,xk} and {y1,…,yk} there are vertex disjoint paths P1,…,Pk such that Pi is directed from xi to yi for i=1,…,k. Pokrovskiy in 2015 proved that every strongly 452k-connected tournament is k-linked. In this paper, we significantly reduce this connectivity bound and show that any (40k−31)-connected tournament is k-linked.

Given a non-trivial finite Abelian group (A,+), let n(A)≥2 be the smallest integer such that for every labelling of the arcs of the bidirected complete graph K↔n(A) with elements from A there exists a directed cycle for which the sum of the arc-labels is zero. The problem of determining n(Zq) for integers q≥2 was recently considered by Alon and Krivelevich, (2020), who proved that n(Zq)=O(qlogq). Here

A regular linear line complex is a three-parameter set of lines in space, whose Plücker vectors lie in a hyperplane, which is not tangent to the Klein quadric. Our main result is a bound O(n1∕2m3∕4+m+n) for the number of incidences between n lines in a complex and m points in F3, where F is a field, and n≤char(F)4∕3 in positive characteristic. Zahl has recently observed that bichromatic pairwise incidences

The descent polynomial of a finite I⊆Z+ is the polynomial d(I,n), for which the evaluation at n>max(I) is the number of permutations on n elements, such that I is the set of indices where the permutation is descending. In this paper, we will prove some conjectures concerning coefficient sequences of d(I,n). As a corollary, we will describe some zero-free regions for the descent polynomial.

Let X,Y be finite sets, r,s,h,λ∈N with s≥r,X⊊Y. By λXh we mean the collection of all h-subsets of X where each subset occurs λ times. A coloring (partition) of λXh is r-regular if each element of X is in exactly r subsets of each color. A one-regular color class is a perfect matching. We are interested in necessary and sufficient conditions under which an r-regular coloring of λXh can be embedded into

The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, K4)-free graphs was constructed, that still has unbounded tree-width (Sintiari and Trotignon, 2019). The class has unbounded degree and contains arbitrarily large clique-minors. We ask whether this is necessary. We prove that for every graph G, if G excludes a fixed

The character theory of symmetric groups, and the theory of symmetric functions, both make use of the combinatorics of Young tableaux, such as the Robinson–Schensted algorithm, Schützenberger’s “jeu de taquin”, and evacuation. In 1995 Poirier and the second author introduced some algebraic structures, different from the plactic monoid, which induce some products and coproducts of tableaux, with homomorphisms

Descent polynomials and peak polynomials, which enumerate permutations π∈Sn with given descent and peak sets respectively, have recently received considerable attention (Billey et al., 1989;Diaz-Lopez et al., 2019). We give several formulas for q-analogs of these polynomials which refine the enumeration by the length of π. In the case of q-descent polynomials we prove that the coefficients in one basis

A subtree of a tree is any induced subgraph that is again a tree (i.e., connected). The mean subtree order of a tree is the average number of vertices of its subtrees. This invariant was first analyzed in the 1980s by Jamison. An intriguing open question raised by Jamison asks whether the maximum of the mean subtree order, given the order of the tree, is always attained by some caterpillar. While we

Although polyhedra can have much fewer edges than triangulations, many results about hamiltonicity proven for triangulations also hold for polyhedra. The most famous of these results is surely Whitney’s result from 1931 that 4-connected triangulations are hamiltonian, which was 25 years later generalised to 4-connected polyhedra by Tutte. Nevertheless the only known bounds for the number of hamiltonian

Let Γ denote a finite, simple, connected and bipartite graph. Fix a vertex x of Γ and let T=T(x) denote the Terwilliger algebra of Γ with respect to x. Assume that x is a distance-regularized vertex, which is not a leaf. We consider the property that Γ has, up to isomorphism, a unique irreducible T-module with endpoint 1, and that this T-module is thin. The main result of the paper is a combinatorial

A fundamental result in extremal set theory is Katona’s shadow intersection theorem, which extends the Kruskal–Katona theorem by giving a lower bound on the size of the shadow of an intersecting family of k-sets in terms of its size. We improve this classical result and a related result of Ahlswede, Aydinian and Khachatrian by proving tight bounds for families that can be quite small. For example,

One may generalize integer compositions by replacing the positive integers with a different additive semigroup, giving the broader concept of a “composition over a semigroup”. Here we focus on semigroups which are finite groups and achieve asymptotic enumeration of compositions over a finite group which satisfy a local restriction. These compositions are associated to walks on a voltage graph whose

We determine all connected {K1,3,K5−e}-free graphs whose second largest eigenvalue does not exceed 1. Our result includes all connected line graphs with the same spectral property, and therefore strengthens the result of Petrović and Milekić (1998).

In this paper, we describe the structure of maximal non-trivial uniform t-intersecting families with large size for finite sets. In the special case when t=1, our result gives rise to Kostochka and Mubayi’s result in 2017.

We show that the cop number of directed and undirected Cayley graphs on abelian groups is in O(n), where n is the number of vertices, by introducing a refined inductive method. With our method, we improve the previous upper bound on cop number for undirected Cayley graphs on abelian groups, and we establish an upper bound on the cop number of directed Cayley graphs on abelian groups. We also use Cayley

In a directed graph D, given two distinct vertices u and v, the line defined by the ordered pair (u,v) is the set of all vertices w such that u,v and w belong to a shortest directed path in D, containing a shortest directed path from u to v. In this work we study the following conjecture: the number of distinct lines in any strongly connected graph is at least its number of vertices, unless there is

In 2006, Chapoton defined a class of Tamari intervals called ”new intervals” in his enumeration of Tamari intervals, and he found that these new intervals are equi-enumerated with bipartite planar maps. We present here a direct bijection between these two classes of objects using a new object called ”degree tree”. Our bijection also gives an intuitive proof of an unpublished equi-distribution result

In this paper, we compute explicitly the q-dimensions of highest weight crystals modulo qn−1 for a quantum group of arbitrary finite type under certain assumption, and interpret the modulo computations in terms of the cyclic sieving phenomenon. This interpretation gives an affirmative answer to the conjecture by Alexandersson and Amini. As an application, under the assumption that λ is a partition

The Jacobian elliptic function sn(u,k) is the inverse of the elliptic integral of the first kind and cn(u,k)=1−sn2(u,k). In this paper, we study coefficient polynomials in the Taylor series expansions of sn(u,k) and cn(u,k). We first provide a combinatorial expansion for a family of bivariate peak polynomials, which count permutations by their odd and even cycle peaks. A special case of this combinatorial

The Pentagon Problem of Erdős problem asks to find an n-vertex triangle-free graph that is maximizing the number of 5-cycles. The problem was solved using flag algebras by Grzesik and independently by Hatami, Hladký, Král’, Norin, and Razborov. Recently, Palmer suggested a more general problem of maximizing the number of 5-cycles in Kk+1-free graphs. Using flag algebras, we show that every Kk+1-free

In this paper we define a new class of partially filled arrays, called λ-fold relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. After showing the connection of this new concept with several other ones, such as signed magic arrays, graph decompositions and relative difference families, we determine some necessary conditions and we present existence

Partial duality is a duality of ribbon graphs relative to a subset of their edges generalizing the classical Euler–Poincaré duality. This operation often changes the genus. Recently J.L. Gross, T. Mansour, and T.W. Tucker formulated a conjecture that for any ribbon graph different from plane trees and their partial duals, there is a subset of edges partial duality relative to which does change the

We prove that for every m∈N and every δ∈(−m,0), the chromatic number of the preferential attachment graph PAt(m,δ) is asymptotically almost surely equal to m+1. The proof relies on a combinatorial construction of a family of digraphs of chromatic number m+1 followed by a proof that asymptotically almost surely there is a digraph in this family, which is realised as a subgraph of the preferential attachment

Conder and the author (Conder and Ma, 2015) studied the orientably-regular maps with simple underlying graphs and showed that there exists at least one orientably-regular map of genus g⁄≡2mod6 with simple underlying graph, and Conder conjectured that there exists at least one for every positive integer g. In this paper, we give a classification of all orientably-regular maps of Euler characteristic

A cellular string of a polytope is a sequence of faces stacked on top of each other in a given direction. The poset of cellular strings, ordered by refinement, is known to be homotopy equivalent to a sphere. The subposet of coherent cellular strings is the face lattice of the fiber polytope, hence is homeomorphic to a sphere. In some special cases, every cellular string is coherent. Such polytopes

We generalize a result of Balister, Győri, Lehel and Schelp for hypergraphs. We determine the unique extremal structure of an n-vertex r-uniform connected hypergraph with the maximum number of hyperedges, without a k-Berge-path, where n≥Nk,r, k≥2r+13>17.

The ball number of a link L, denoted by ball(L), is the minimum number of solid balls (not necessarily of the same size) needed to realize a necklace representing L. In this paper, we show that ball(L)≤5cr(L) where cr(L) denotes the crossing number of a nontrivial nonsplittable link L. To this end, we use the connection of the Lorentz geometry with the ball packings. The well-known Koebe–Andreev–Thurston

It is shown that each subgroup of odd index in an alternating group of degree at least 10 has all insoluble composition factors to be alternating. A classification is then given of 2-arc-transitive graphs of odd order admitting an alternating group or a symmetric group. This is the second of a series of papers aiming towards a classification of 2-arc-transitive graphs of odd order.

A coloring of edges of a graph G is injective if for any two distinct edges e1 and e2, the colors of e1 and e2 are distinct if they are at distance 1 in G or in a common triangle. Naturally, the injective chromatic index of G, χinj′(G), is the minimum number of colors needed for an injective edge-coloring of G. We study how large can be the injective chromatic index of G in terms of maximum degree

Consider the following game played by two players, called Waiter and Client, on the edges of Kn (where n is divisible by 3). Initially, all the edges are unclaimed. In each round, Waiter picks two yet unclaimed edges. Client then chooses one of these two edges to be added to Waiter’s graph and one to be added to Client’s graph. Waiter wins if she forces Client to create a K3-factor in Client’s graph

Suppose I and J are proper ideals on some set X. We say that I and J are incompatible if I∪J does not generate a proper ideal. Equivalently, I and J are incompatible if there is some A⊆X such that A∈I and X∖A∈J. If some B⊆X is either in I∖J or in J∖I, then we say that B chooses between I and J. We consider the following Ramsey-theoretic problem: Given several pairs (I1,J1),(I2,J2),…,(Ik,Jk) of incompatible

We present new short proofs to both the exact and the stability result of two extremal problems. The first result is about the extension of Turán’s theorem to hypergraphs, and the second result is about cancellative hypergraphs. Our proofs are concise and straightforward, but give a sharper version of stability theorems to both problems.

Characterizing graphs by their spectra is an important topic in spectral graph theory, which has attracted a lot of attention of researchers in recent years. It is generally very hard and challenging to show a given graph is determined by its spectrum. In Wang (2017), the author gave a simple arithmetic condition for a family of graphs being determined by their generalized spectra. However, the method

A well-known result of Bollobás says that if {(Ai,Bi)}i=1m is a set pair system such that |Ai|≤a and |Bi|≤b for 1≤i≤m, and Ai∩Bj≠0̸ if and only if i≠j, then m≤a+ba. Füredi, Gyárfás and Király recently initiated the study of such systems with the additional property that |Ai∩Bj|=1 for all i≠j. Confirming a conjecture of theirs, we show that this extra condition allows an improvement of the upper bound

The fractional and circular chromatic numbers are the two most studied non-integral refinements of the chromatic number of a graph. Starting from the definition of a coloring base of a graph, which originated in work related to ergodic theory, we formalize the notion of a gyrocoloring of a graph: the vertices are colored by translates of a single Borel set in the circle group, and neighboring vertices

For a given graph H, the Ramsey number r(H) is the minimum N such that any 2-edge-coloring of the complete graph KN yields a monochromatic copy of H. Given a positive integer n, a fanFn is a graph formed by n triangles that share one common vertex. We show that 9n∕2−5≤r(Fn)≤11n∕2+6 for any n. This improves previous best bounds r(Fn)≤6n of Lin and Li and r(Fn)≥4n+2 of Zhang, Broersma and Chen.

It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. This conjecture has been verified for many families of snarks with small (≤5) cyclic edge-connectivity. An infinite family, denoted by SK, of cyclically 6-edge-connected superposition snarks was constructed in [European J. Combin. 2002] by Kochol

The partial (Poincaré) dual with respect to a subset A of edges of a ribbon graph G was introduced by Chmutov in connection with the Jones–Kauffman and Bollobás–Riordan polynomials. In developing the theory of maps, Wilson and others have composed Poincaré duality ∗ with Petrie duality × to give Wilson duality ∗×∗ and two trialities ∗× and ×∗. In further expanding the theory, Abrams and Ellis-Monaghan

The present paper is to honour Gyula Katona, my teacher on the occasion of his 80th birthday. Its main content is threefold, new proofs of old results (e.g. the De Bonis, Katona, Swanepoel Theorem on butterfly-free families), new results obtained via the Katona Circle (e.g. for the sum of the sizes of non-empty cross-intersecting families), the solution for the Katona Circle of some notoriously difficult