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

A family F⊂[n]k is U(s,q) of for any F1,…,Fs∈F we have |F1∪⋯∪Fs|≤q. This notion generalizes the property of a family to be t-intersecting and to have matching number smaller than s. In this paper, we find the maximum |F| for F that are U(s,q), provided n>C(s,q)k with moderate C(s,q). In particular, we generalize the result of the first author on the Erdős Matching Conjecture and prove a generalization

We introduce the hydra continued fractions, as a generalization of the Rogers–Ramanujan continued fractions in the context of noncommutative series, and give them a combinatorial interpretation in terms of shift-plethystic trees. We show it is possible to express an m−1 headed hydra continued fraction as a quotient of m-distinct partition generating functions, and in its dual form as a quotient of

We present a new definition of non-ambiguous trees (NATs) as labelled binary trees. We thus get a differential equation whose solution can be described combinatorially. This yields a new formula for the number of NATs. We also obtain q-versions of our formula. We finally generalise NATs to higher dimension.

A graph puzzle Puz(G) of a graph G is defined as follows. A configuration of Puz(G) is a bijection from the set of vertices of a board graph to the set of vertices of a pebble graph, both graphs being isomorphic to some input graph G. A move of pebbles is defined as exchanging two pebbles which are adjacent on both a board graph and a pebble graph. For a pair of configurations f and g, we say that

We apply a one-dimensional discrete dynamical system originally considered by Arnol’d reminiscent of mathematical billiards to the study of two-move riders, a type of fairy chess piece. In this model, particles travel through a bounded convex region along line segments of one of two fixed slopes. We apply this dynamical system to characterize the vertices of the inside-out polytope arising from counting

Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with n vertices is (1+o(1))3n22π2. This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap (algebraic connectivity) of a connected k-regular graph on n vertices is at least (1+o(1))2kπ23n2, and the bound is attained for at least one value of k. Based upon

We derive a general upper bound for the number of incidences with k-dimensional varieties in Rd. The leading term of this new bound generalizes previous bounds for the special cases of k=1,k=d−1, and k=d∕2, to every 1≤k

We construct a countable planar graph which, for any two vertices u,v and any integer k≥1, contains k edge-disjoint order-compatible u–v paths but not infinitely many. This graph has applications in Ramsey theory, in the study of connectivity and in the characterisation of the Farey graph.

In a proper edge-coloring of a cubic graph, an edge e is normal if the set of colors used by the five edges incident with an end of e has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless cubic graph has a normal 5-edge-coloring, that is, a proper 5-edge-coloring such that all edges are normal. In this paper, we prove a result related to the Petersen coloring conjecture

We give a new characterization of biinfinite Sturmian sequences in terms of indistinguishable asymptotic pairs. Two asymptotic sequences on a full Z-shift are indistinguishable if the sets of occurrences of every pattern in each sequence coincide up to a finitely supported permutation. This characterization can be seen as an extension to biinfinite sequences of Pirillo’s theorem which characterizes

In this paper we investigate algebraic properties of big Ramsey degrees in categories satisfying some mild conditions. As the first nontrivial consequence of the generalization we advocate in this paper we prove that small Ramsey degrees are the minima of the corresponding big ones. We also prove that big Ramsey degrees are subadditive and show that equality is enforced by an abstract property of objects

We consider edge decompositions of the n-dimensional hypercube Qn into isomorphic copies of a given graph H. While a number of results are known about decomposing Qn into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if n is even, ℓ<2n and ℓ divides the number of edges of Qn, then the path of length

Nikiforov (2002) showed that if G is Kr+1-free then the spectral radius ρ(G)≤2m(1−1∕r), which implies that G contains C3 if ρ(G)>m. In this paper, we follow this direction in determining which subgraphs will be contained in G if ρ(G)>f(m), where f(m)∼m as m→∞. We first show that if ρ(G)≥m, then G contains K2,r+1 unless G is a star; and G contains either C3+ or C4+ unless G is a complete bipartite graph

The eternal vertex colouring game, recently introduced byKlostermeyer and Mendoza (2018), is a version of the vertex colouring game, a well studied graph game. The game is played by two players, Alice and Bob on a graph G with a set of colours {1,…,k}. The game consists of rounds, such that in each round, every vertex is coloured exactly once. In the first round, players alternate turns with Alice

In 1964, Erdős, Hajnal and Moon introduced a saturation version of Turán’s classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a Kr-free, n-vertex graph with the property that the addition of any further edge yields a copy of Kr. We consider analogues of this problem in other settings. We prove a saturation version of the Erdős–Szekeres theorem

The reconfiguration graph Rk(G) for the k-colorings of a graph G has as vertices all possible k-colorings of G and two colorings are adjacent if they differ in the color of exactly one vertex. A result of Bousquet and Perarnau (2016) regarding graphs of bounded degeneracy implies that for a planar graph G with n vertices, R12(G) has diameter at most 6n, and if G is triangle-free, then R8(G) has diameter

The metric dimension of a graph G is the minimal size of a subset R of vertices of G that, upon reporting their graph distance from a distinguished (source) vertex v⋆, enable unique identification of the source vertex v⋆ among all possible vertices of G. In this paper we show a Law of Large Numbers (LLN) for the metric dimension of some classes of trees: critical Galton–Watson trees conditioned to

We present a bijection for toroidal maps that are essentially 3-connected (3-connected in the periodic planar representation). Our construction actually proceeds on certain closely related bipartite toroidal maps with all faces of degree 4 except for a hexagonal root-face. We show that these maps are in bijection with certain well-characterized bipartite unicellular maps. Our bijection, closely related

We consider union-closed set systems with infinite breadth, focusing on three particular configurations Tmax(E), Tmin(E) and Tort(E). We show that these three configurations are not isolated examples; in any given union-closed set system of infinite breadth, at least one of these three configurations will occur as a subprojection. This characterizes those union-closed set systems which have infinite

A graph with a semiregular group of automorphisms can be thought of as the derived cover arising from a voltage graph. Since its inception, the theory of voltage graphs and their derived covers has been a powerful tool used in the study of graphs with a significant degree of symmetry. We generalise this theory to graphs with a group of automorphisms that is not necessarily semiregular, and we generalise

A blocking set in a projective plane is a point set intersecting each line. The smallest blocking sets are lines. The second smallest minimal blocking sets are Baer subplanes (subplanes of order q). Our aim is to study the stability of Baer subplanes in PG(2,q). If we delete q+1−k points from a Baer subplane, then the resulting set has size q+k and (q+1−k)(q−q) 0-secants. If we have somewhat more 0-secants

Given a group G of automorphisms of a matroid M, we describe the representations of G on the homology of the independence complex of the dual matroid M∗. These representations are related to the homology of the lattice of flats of M, and (when M is realizable) to the top cohomology of a hyperplane arrangement. Finally, we analyze in detail the case of the complete graph, which has applications to algebraic

For a positive integer r, a distance-r independent set in an undirected graph G is a set I⊆V(G) of vertices pairwise at distance greater than r, while a distance-r dominating set is a set D⊆V(G) such that every vertex of the graph is within distance at most r from a vertex from D. We study the duality between the maximum size of a distance-2r independent set and the minimum size of a distance-r dominating

For lattice paths in strips which begin at (0,0) and have only up steps U:(i,j)→(i+1,j+1) and down steps D:(i,j)→(i+1,j−1), let An,k denote the set of paths of length n which start at (0,0), end on heights 0 or −1, and are contained in the strip −⌊k+12⌋≤y≤⌊k2⌋ of width k, and let Bn,k denote the set of paths of length n which start at (0,0) and are contained in the strip 0≤y≤k. We establish a bijection

Adiprasito and Nevo (2018) proved that there exists a set of 76 points in R3 such that every triangulated planar graph has an infinitesimally rigid realization in which each vertex is mapped to a point in this set. In this paper we show that there exists a set of 26 points in the plane such that every planar graph which is generically rigid in R2 has an infinitesimally rigid realization in which each

The well-known 1–2–3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with 1, 2 and 3 so that adjacent vertices receive distinct weighted degrees. This is open in general. The Standard (2,2)-Conjecture asserts that every graph with no isolated edge and no isolated triangle can be decomposed into two graphs, each of which can be weighted with 1, 2 for the same

In 1965, Paul Erdős asked about the largest family Y of k-sets in {1,…,n} such that Y does not contain s+1 pairwise disjoint sets. This problem is commonly known as the Erdős Matching Conjecture. We investigate the q-analog of this question, that is we want to determine the size of a largest family Y of k-spaces in Fqn such that Y does not contain s+1 pairwise disjoint k-spaces. Here we call two subspaces

There is a natural analogue of weak Bruhat order on the involutions in any Coxeter group. The saturated chains of intervals in this order correspond to reduced words for a certain set of group elements called atoms. Brion gives a general formula for the cohomology class of a K-orbit closure in an arbitrary flag variety, where K is a symmetric subgroup of a complex algebraic group. In type A, the terms

The entire chromatic number χvef(G) of a plane graph G is the least number of colors such that any two adjacent or incident elements in V(G)∪E(G)∪F(G) receive different colors. In this paper, we prove that every 2-connected simple plane graph G with maximum degree Δ≥20 has χvef(G)=Δ+1.

Plummer and Toft conjectured in 1987 that the vertices of every 3-connected plane graph with maximum face size Δ⋆ can be colored using at most Δ⋆+2 colors in such a way that no face is incident with two vertices of the same color. The conjecture has been proven for Δ⋆=3, Δ⋆=4 and Δ⋆≥18. We prove the conjecture for Δ⋆=16 and Δ⋆=17.

We attempt to generalize a theorem of Nash-Williams stating that a graph has a k-arc-connected orientation if and only if it is 2k-edge-connected. In a strongly connected digraph we call an arc deletable if its deletion leaves a strongly connected digraph. Given a 3-edge-connected graph G, we define its Frank number f(G) to be the minimum number k such that there exist k orientations of G with the

Let N be the set of all nonnegative integers. For S⊆N and n∈N, let RS(n) denote the number of solutions of the equation n=s+s′, s,s′∈S, s

In this paper, we consider various classes of polyiamonds that are animals residing on the triangular lattice. By careful analyses through certain layer-by-layer decompositions and cell pruning/growing arguments, we derive explicit forms for the generating functions of the number of nonempty translation-invariant baryiamonds (bargraphs in the triangular lattice), column-convex polyiamonds, and convex

Building on previous work of Di Nasso and Luperi Baglini, we provide general necessary conditions for a Diophantine equation to be partition regular. These conditions are inspired by Rado’s characterization of partition regular linear homogeneous equations. We conjecture that these conditions are also sufficient for partition regularity, at least for equations whose corresponding monovariate polynomial

The principal specialization νw=Sw(1,…,1) of the Schubert polynomial at w, which equals the degree of the matrix Schubert variety corresponding to w, has attracted a lot of attention in recent years. In this paper, we show that νw is bounded below by 1+p132(w)+p1432(w) where pu(w) is the number of occurrences of the pattern u in w, strengthening a previous result by A. Weigandt. We then make a conjecture

Recently Okada defined algebraically ninth variation skew Q-functions, in parallel to Macdonald’s ninth variation skew Schur functions. Here we introduce a skew shifted tableaux definition of these ninth variation skew Q-functions, and prove by means of a non-intersecting lattice path model a Pfaffian outside decomposition result in the form of a ninth variation version of Hamel’s Pfaffian outside

A certain signed adjacency matrix of the hypercube, which Hao Huang used last year to resolve the Sensitivity Conjecture, is closely related to the unique, 4-cycle free, 2-fold cover of the hypercube. We develop a framework in which this connection is a natural first example of the relationship between group valued adjacency matrices with few eigenvalues, and combinatorially interesting covering graphs

An r-uniform hypergraph is linear if every two edges intersect in at most one vertex. Given a family of r-uniform hypergraphs F, the linear Turán number exrlin(n,F) is the maximum number of edges of a linear r-uniform hypergraph on n vertices that does not contain any member of F as a subhypergraph. Given a graph F and a positive integer r≥2, the r-expansion of F is the r-graph F+ obtained from F by

Recently, Yan and the first author investigated systematically the enumeration of inversion or ascent sequences avoiding vincular patterns of length 3, where two of the three letters are required to be adjacent. They established many connections with familiar combinatorial families and proposed several interesting conjectures. The main objective of this paper is to prove two of their conjectures concerning

DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvořák and Postle. We introduce and study (i,j)-defective DP-colorings of multigraphs. We concentrate on sparse multigraphs and consider fDP(i,j,n) — the minimum number of edges that may have an n-vertex (i,j)-critical multigraph, that is, a multigraph G that has no (i,j)-defective DP-coloring

We introduce a counterpart to the notion of tilings, that is vertex-disjoint copies of a fixed graph F, to the setting of graphons. The case F=K2 gives the notion of matchings in graphons. We give a transference statement that allows us to switch between the finite and limit notion, and derive several favorable properties, including the LP-duality counterpart to the classical relation between the fractional

In this paper, we study some Euler–Apéry-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple zeta values. Based on these formulas, we further show that some other series are reducible to ln(2), zeta values, and alternating multiple zeta values by

Let G be a simple graph on n vertices. Let LG and IG denote the Lovász–Saks–Schrijver(LSS) ideal and parity binomial edge ideal of G in the polynomial ring S=K[x1,…,xn,y1,…,yn] respectively. We classify graphs whose LSS ideals and parity binomial edge ideals are complete intersections. We also classify graphs whose LSS ideals and parity binomial edge ideals are almost complete intersections, and we

In this paper, we introduce a new surgery operation on the set of collections of curves, on a closed oriented surface, whose complement is a topological disk. We prove that any two such collections can be connected by a sequence of surgeries.

Let Dn denote the average number of iterations of West’s stack-sorting map s that are needed to sort a permutation in Sn into the identity permutation 123⋯n. We prove that 0.62433≈λ≤lim infn→∞Dnn≤lim supn→∞Dnn≤35(7−8log2)≈0.87289, where λ is the Golomb–Dickman constant. Our lower bound improves upon West’s lower bound of 0.23, and our upper bound is the first improvement upon the trivial upper bound

Let n>k>0 be integers and X an n-element set. A family F consisting of subsets of X is called k-Sperner if it has no distinct members F0,…,Fk such that F0⊂F1⊂…⊂Fk. A family is called s-union if the union of any two of its members has size at most s. A classical result of Milner determines the maximum size of a family that is both 1-Sperner and s-union. The present paper is dealing with the case k≥2