We prove the following conjecture of S. Thomassé: for every (potentially infinite) digraph D it is possible to iteratively reverse directed cycles in such a way that the dichromatic number of the final reorientation D∗ of D is at most two and each edge is reversed only finitely many times. In addition, we guarantee that in every strong component of D∗ all the local edge-connectivities are finite and

We consider the problem of determining when the difference of two ribbon Schur functions is a single Schur function. We fully classify the five infinite families of pairs of ribbon Schur functions whose difference is a single Schur function with corresponding partition having at most two parts at least 2. We also prove an identity for differences of ribbon Schur functions and we determine some necessary

A set A of natural numbers possesses property Ph, if there are no distinct elements a0,a1,…,ah∈A with a0 dividing the product a1a2…ah. Erdős determined the maximum size of a subset of {1,…,n} possessing property P2. More recently, Chan et al. (2010) solved the case h=3, finally the general case also got resolved by Chan (2011), the maximum size is π(n)+Θh(n2∕(h+1)(logn)2). In this note we consider

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph G=(V,E) and a subset A of E we let λG(A) be the number of vertices incident with an edge in A and an edge in E∖A. For a subset X of V, let

In this paper we prove, in a purely combinatorial-algebraic way, a structural quasi-polynomiality property for the Bousquet-Mélou–Schaeffer numbers. Conjecturally, this property should follow from the Chekhov–Eynard–Orantin topological recursion for these numbers (or, to be more precise, the Bouchard–Eynard version of the topological recursion for higher order critical points), which we derive in this

The cut-rank of a set X of vertices in a graph G is defined as the rank of the X×(V(G)∖X) matrix over the binary field whose (i,j)-entry is 1 if the vertex i in X is adjacent to the vertex j in V(G)∖X and 0 otherwise. We introduce the graph parameter called the average cut-rank of a graph, defined as the expected value of the cut-rank of a random set of vertices. We show that this parameter does not

An (n,k)-Sperner partition system is a collection of partitions of some n-set, each into k nonempty classes, such that no class of any partition is a subset of a class of any other. The maximum number of partitions in an (n,k)-Sperner partition system is denoted SP(n,k). In this paper we introduce a new construction for Sperner partition systems and use it to asymptotically determine SP(n,k) in many

This paper considers two important questions in the well-studied theory of graphs that are F-saturated. A graph G is called F-saturated if G does not contain a subgraph isomorphic to F, but the addition of any edge creates a copy of F. We first resolve a fundamental question of minimizing the number of cliques of size r in a Ks-saturated graph for all sufficiently large numbers of vertices, confirming

We show that Γ-free fillings and lonesum fillings of Ferrers shapes are equinumerous by applying a previously defined bijection on matrices for this more general case and by constructing a new bijection between Callan sequences and Dumont-like permutations. As an application, we give a new combinatorial interpretation of Genocchi numbers in terms of Callan sequences.Further, we recover some of Hetyei’s

For a tree T, the subtree polynomial of T is the generating polynomial for the number of subtrees of T. We show that the complex roots of the subtree polynomial are contained in the disk z∈ℂ:|z|≤1+33, and that K1,3 is the only tree whose subtree polynomial has a root on the boundary. We also prove that the closure of the collection of all real roots of subtree polynomials contains the interval [−2

We introduce ideas that complement the many known connections between polymatroids and graph coloring. Given a hypergraph that satisfies certain conditions, we construct polymatroids, given as rank functions, that can be written as sums of rank functions of matroids, and for which the minimum number of matroids required in such sums is the chromatic number of the line graph of the hypergraph. This

We introduce two classes of morphisms over the alphabet A={0,1} whose fixed points contain infinitely many antipalindromic factors. An antipalindrome is a finite word invariant under the action of the antimorphism E:{0,1}∗→{0,1}∗, defined by E(w1⋯wn)=(1−wn)⋯(1−w1). We conjecture that these two classes contain all morphisms (up to conjugation) which generate infinite words with infinitely many antipalindromes

We introduce patterns on a triangular grid generated by paperfolding operations. We show that in case these patterns are defined using a periodic sequence of foldings, they can also be generated using substitution rules and compute eigenvalues and eigenvectors of the corresponding matrices. We also prove that densities of all basic triangles are equal in these patterns.

Let S be a set of arbitrary objects, and let Sd={v1...vd:vi∈S}. A polybox code is a set V⊂Sd with the property that for every two words v,w∈V there is i∈[d] with vi′=wi, where a permutation s↦s′ of S is such that s′′=(s′)′=s and s′≠s. If |V|=2d, then V is called a cube tiling code. Cube tiling codes determine 2-periodic cube tilings of Rd or, equivalently, tilings of the flat torus Td={(x1,…,xd)(mod2):(x1

Esperet, de Joannis de Verclos, Le and Thomassé in [SIAM J. Discrete Math., 32(1) (2018), 534–542] introduced the problem that for an odd prime p, whether there exists an orientation D of a graph G for any mapping f:E(G)→Zp∗ and any Zp-boundary b of G, such that under D, at every vertex, the net out f-flow is the same as b(v) in Zp. Such an orientation D is called an (f,b;p)-orientation of G. Esperet

We study the sum–product problem for the planar hypercomplex numbers: the dual numbers and double numbers. These number systems are similar to the complex numbers, but it turns out that they have a very different combinatorial behavior. We identify parameters that control the behavior of these problems, and derive sum–product bounds that depend on these parameters. For the dual numbers we expose a

We study the flow extension of graphs, i.e., pre-assigning a partial flow on the edges incident to a given vertex and aiming to extend to the entire graph. This is closely related to Tutte’s 3-flow conjecture(1972) that every 4-edge-connected graph admits a nowhere-zero 3-flow and a Z3-group connectivity conjecture(3GCC) of Jaeger, Linial, Payan, and Tarsi(1992) that every 5-edge-connected graph G

Recently, several hypergraph Turán problems were solved by the powerful random algebraic method. However, the random algebraic method usually requires some parameters to be very large, hence we are concerned about how these Turán numbers depend on such large parameters of the forbidden hypergraphs. In this paper, we determine the dependence on such specified large constant for several hypergraph Turán

Let po(n) denote the number of partitions of n with more odd parts than even parts and let pe(n) denote the number of partitions of n with more even parts than odd parts. Using q-series transformations we find a generating function for po(n)−pe(n), which implies that po(n)>pe(n) for all positive integers n≠2. Using combinatorial mappings we prove a stronger result, namely that for all n>7 we have 2pe(n)

In this paper, we consider the question of when a strongly regular graph with parameters ((s+1)(st+1),s(t+1),s−1,t+1) can exist. A strongly regular graph with such parameters is called a pseudo-generalized quadrangle. A pseudo-generalized quadrangle can be derived from a generalized quadrangle, but there are other examples which do not arise in this manner. If the graph is derived from a generalized

We study Erdős–Szekeres-type problems for k-convex point sets, a recently introduced notion that naturally extends the concept of convex position. A finite set S of n points is k-convex if there exists a spanning simple polygonization of S such that the intersection of any straight line with its interior consists of at most k connected components. We address several open problems about k-convex point

For a sequence A of positive integers, let P(A) be the set of all integers which can be represented as the finite sum of distinct terms of A. In 2012, Chen and Fang proved that, for a sequence of integers B={b1

We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, for various models of random (multi)graphs. For our proofs we introduce the notion of patchworks to describe the possible overlappings of copies of subgraphs. Furthermore, the proofs are based on analytic combinatorics to carry out asymptotic computations. The flexibility of our approach allows

An edge-colouring of a graph is distinguishing if the only automorphism that preserves the colouring is the identity. It has been conjectured that all but finitely many connected, finite, regular graphs admit a distinguishing edge-colouring with two colours. We show that all such graphs except K2 admit a distinguishing edge-colouring with three colours. This result also extends to infinite, locally

We study the positional game where two players, Maker and Breaker, alternately select respectively 1 and b previously unclaimed edges of Kn. Maker wins if she succeeds in claiming all edges of some odd cycle in Kn and Breaker wins otherwise. Improving on a result of Bednarska and Pikhurko, we show that Maker wins the odd cycle game if b≤((4−6)∕5+o(1))n. We furthermore introduce “connected rules” and

The notion of homomorphism homogeneity was introduced by Cameron and Nešetřil as a natural generalization of the classical model-theoretic notion of homogeneity. A relational structure is called homomorphism homogeneous (HH) if every homomorphism between finite substructures extends to an endomorphism. It is called polymorphism homogeneous (PH) if every finite power of the structure is homomorphism

The existence of Steiner triple systems STS(n) of order n containing no nontrivial subsystem is well known for every admissible n. We generalize this result in two ways. First we define the expander property of 3-uniform hypergraphs and show the existence of Steiner triple systems which are almost perfect expanders. Next we define the strong and weak spreading properties of linear hypergraphs, and

A Cayley graph of a group H is a finite simple graph Γ such that its automorphism group Aut(Γ) contains a subgroup isomorphic to H acting regularly on V(Γ), while a Haar graph of H is a finite simple bipartite graph Σ such that Aut(Σ) contains a subgroup isomorphic to H acting semiregularly on V(Σ) and the H-orbits are equal to the partite sets of Σ. It is well-known that every Haar graph of finite

We study the class L of link-types that admit a K4-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K4. We prove that L is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate L (and subclasses of it), with respect to the minimum number of crossings or edges

For integers n≥k≥2, let V be an n-element set, and let Vk denote the set of all k-element subsets of V. For disjoint A,B⊆V, we say {A,B} covers K∈Vk if K⊆A∪̇B and K meets each of A and B, i.e., K∩A≠0̸≠K∩B. We say that a collection C of such pairs {A,B} covers Vk if every element of Vk is covered by at least one member of C. When k=2, such a family is called a separating system of V, where this concept

We present counterexamples to a 30-year-old conjecture of Las Vergnas (1988) regarding the Tutte polynomial of binary matroids.

We study the row-space partition and the pivot partition on the matrix space Fqn×m. We show that both these partitions are reflexive and that the row-space partition is self-dual. Moreover, using various combinatorial methods, we explicitly compute the Krawtchouk coefficients associated with these partitions. This establishes MacWilliams-type identities for the row-space and pivot enumerators of linear

If a graph G has distinguishing number 2, then there exists a partition of its vertex set into two parts, such that no nontrivial automorphism of G fixes setwise the two parts. Such a partition is called a 2-distinguishing coloring of G, and the parts are called its color classes. If G admits such a coloring, it is often possible to find another in which one of the color classes is sparse in a certain

In this paper we consider the question of the existence of Hamiltonian circuits in the tope graphs of central arrangements of hyperplanes. Some of the results describe connections between the existence of Hamiltonian circuits in the arrangement and the odd-even invariant of the arrangement. In conjunction with this, we present some results concerning bounds on the odd-even invariant. The results given

We study the joint distribution of the input sum and the output sum of a deterministic transducer. Here, the input of this finite-state machine is a uniformly distributed random sequence. We give a simple combinatorial characterization of transducers for which the output sum has bounded variance, and we also provide algebraic and combinatorial characterizations of transducers for which the covariance

Let [Formula: see text] be a Krull monoid with finite class group [Formula: see text] such that every class contains a prime divisor and let [Formula: see text] be the Davenport constant of [Formula: see text]. Then a product of two atoms of [Formula: see text] can be written as a product of at most [Formula: see text] atoms. We study this extremal case and consider the set [Formula: see text] defined

Embedded trees are labelled rooted trees, where the root has zero label and where the labels of adjacent vertices differ (at most) by [Formula: see text]. Recently it has been proved (see Chassaing and Schaeffer (2004) [8] and Janson and Marckert (2005) [11]) that the distribution of the maximum and minimum labels are closely related to the support of the density of the integrated superbrownian excursion

Let [Formula: see text] be a finitely generated group, [Formula: see text] a finite set of generators and [Formula: see text] a subgroup of [Formula: see text]. We define what it means for [Formula: see text] to be a context-free pair; when [Formula: see text] is trivial, this specializes to the standard definition of [Formula: see text] to be a context-free group. We derive some basic properties of

An old conjecture of Graham stated that if [Formula: see text] is a prime and [Formula: see text] is a sequence of [Formula: see text] terms from the cyclic group [Formula: see text] such that all (nontrivial) zero-sum subsequences have the same length, then [Formula: see text] must contain at most two distinct terms. In 1976, Erdős and Szemerédi gave a proof of the conjecture for sufficiently large

We study unfair permutations, which are generated by letting [Formula: see text] players draw numbers and assuming that player [Formula: see text] draws [Formula: see text] times from the unit interval and records her largest value. This model is natural in the context of partitions: the score of the [Formula: see text]th player corresponds to the multiplicity of the summand [Formula: see text] in

Bipartitional relations were introduced by Foata and Zeilberger in their characterization of relations which give rise to equidistribution of the associated inversion statistic and major index. We consider the natural partial order on bipartitional relations given by inclusion. We show that, with respect to this partial order, the bipartitional relations on a set of size [Formula: see text] form a