A d-regular graph on n vertices with the second largest absolute eigenvalue at most λ is called an (n,d,λ)-graph. The celebrated expander mixing lemma for (n,d,λ)-graphs builds a connection between graph spectrum and edge distribution. In this paper, we present some applications of the expander mixing lemma. In particular, we make progress toward the toughness conjecture of Brouwer. The toughness t(G)

There are finitely many simplicial complexes (up to isomorphism) with a given number of vertices. Translating this fact to the language of h-vectors, there are finitely many simplicial complexes of bounded dimension with h1=k for any natural number k. In this paper we study the question at the other end of the h-vector: Are there only finitely many (d−1)-dimensional simplicial complexes with hd=k for

An ordered hypergraph is a hypergraph H with a specified linear ordering of the vertices, and the appearance of an ordered hypergraph G in H must respect the specified order on V(G). In on-line Ramsey theory, Builder iteratively presents edges that Painter must immediately color. The t-color on-line size Ramsey number R̃t(G) of an ordered hypergraph G is the minimum number of edges Builder needs to

A graph is ℓ-reconstructible if it is determined by its multiset of induced subgraphs obtained by deleting ℓ vertices. We prove that 3-regular graphs are 2-reconstructible.

Let r,k≥1 be two integers. An r-hued k-coloring of the vertices of a graph G=(V,E) is a proper k-coloring of the vertices, such that, for every vertex v∈V, the number of colors in its neighborhood is at least min{dG(v),r}, where dG(v) is the degree of v. We prove the existence of an r-hued (r+1)-coloring for planar graphs with girth at least 8 for r≥9. As a corollary, every planar graph with maximum

Let n≥2k−1>1, [n]={1,2,…,n}. For a family F of k-subsets of [n] let ∂F be the immediate shadow (cf. Definition 1.1) of F. Suppose that |F∩F′|≥2 for all F,F′∈F. We conjecture that |F|+|∂F|≤3n−2k−2+n−2k−3 and prove it for n=2k−1, n≥3(k−1) and also for k≤10. This problem is somewhat unusual but we exhibit deep connections to the Erdős–Ko–Rado Theorem and to the Erdős Matching Conjecture. Some related

We introduce the generic Lah polynomials Ln,k(ϕ), which enumerate unordered forests of increasing ordered trees with a weight ϕi for each vertex with i children. We show that, if the weight sequence ϕ is Toeplitz-totally positive, then the triangular array of generic Lah polynomials is totally positive and the sequence of row-generating polynomials Ln(ϕ,y) is coefficientwise Hankel-totally positive

Let Q(n) denote the count of the primitive subsets of the integers {1,2…,n}. We give a new proof that Q(n)=α(1+o(1))n for some constant α, which allows us to give a good error term and to improve upon the lower bound for the value of α. We also show that the method developed can be applied to many similar problems related to the divisor graph, including other questions about primitive sets, geomet

Let Γ be a finite, undirected, connected, simple graph. We say that a matching M is a permutable m-matching if M contains m edges and the subgroup of Aut(Γ) that fixes the matching M setwise allows the edges of M to be permuted in any fashion. A matching M is 2-transitive if the setwise stabilizer of M in Aut(Γ) can map any ordered pair of distinct edges of M to any other ordered pair of distinct edges

We study a generalisation of the bipartite Ramsey numbers to blowups of graphs. For a graph G, denote the t-blowup of G by G[t]. We say that G is r-Ramsey for H, and write G→rH, if every r-colouring of the edges of G has a monochromatic copy of H. We show that if G→rH, then for all t, there exists n such that G[n]→rH[t]. In fact, we provide exponential lower and upper bounds for the minimum n with

The bivariate Eulerian polynomials are defined by An(p,q)=∑π∈Snpodes(π)qedes(π),where odes(π) and edes(π) are the number of descents of permutation π in odd and even positions, respectively. In this paper, by the Hetyei–Reiner action, we show that for k≥1, the bivariate Eulerian polynomials A2k+1(p,q) and (1+p)−1A2k(p,q) are γ-positive, namely, they can be expressed in terms of the basis Bn≔{(pq)i

A relationship between signed Eulerian polynomials and the classical Eulerian polynomials on Sn was given by Désarménien and Foata in 1992, and a refined version, called signed Euler–Mahonian identity, together with a bijective proof was proposed by Wachs in the same year. By generalizing this bijection, in this paper we extend the above results to the Coxeter groups of types Bn, Dn, and the complex

Suppose G is a graph and k,k′ are positive integers. A (k,k′)-list assignment is a mapping L which assigns to each vertex v a set L(v) of k real numbers, and assigns to each edge e a set L(e) of k′ real numbers. A proper L-total weighting is a mapping ϕ:V(G)∪E(G)→R such that ϕ(z)∈L(z) for z∈V∪E, and ∑e∈E(u)ϕ(e)+ϕ(u)≠∑e∈E(v)ϕ(e)+ϕ(v) for every edge uv. A graph G is called (k,k′)-choosable if for every

Circular-perfect graphs form a natural superclass of perfect graphs, introduced by Zhu almost 20 years ago: on the one hand due to their definition by means of a more general coloring concept, on the other hand as an important class of χ-bound graphs with the smallest non-trivial χ-binding function χ(G)≤ω(G)+1. In this paper, we survey the results about circular-perfect graphs obtained in the two last

Máčajová et al. (2016) defined the chromatic number of a signed graph which coincides for all-positive signed graphs with the chromatic number of unsigned graphs. They conjectured that in this setting, for signed planar graphs four colors are always enough, generalizing thereby The Four Color Theorem. We disprove the conjecture.

In 1964 Erdős proved that (1+o1)eln(2)4k22k edges are sufficient to build a k-graph which is not two colorable. To this day, it is not known whether there exist such k-graphs with smaller number of edges. Erdős’ bound is consequence of the fact that a hypergraph with k2∕2 vertices and M(k)=(1+o1)eln(2)4k22k randomly chosen edges of size k is asymptotically almost surely not two colorable. Our first

In this paper we propose a new approach to superposition of snarks, a powerful method of constructing large cubic graphs with no 3-edge-colouring from small ones. The main idea is to use surjective mappings between graphs similar to graph homomorphisms and to control flows induced from the domain graph to the target graph via the mappings. This leads to significant strengthening of the power of the

A signed graph is a graph together with an assignment of signs to the edges. A closed walk in a signed graph is said to be positive (negative) if it has an even (odd) number of negative edges, counting repetition. Recognizing the signs of closed walks as one of the key structural properties of a signed graph, we define a homomorphism of a signed graph (G,σ) to a signed graph (H,π) to be a mapping of

In 1999, at one of his last public lectures, Tutte discussed a question he had considered since the times of the Four Color Conjecture. He asked whether the 4-coloring complex of a planar triangulation could have two components in which all colorings had the same parity. In this note we answer Tutte’s question contrary to his speculations by showing that there are triangulations of the plane whose

Let G be a graph, and let k be a positive integer. The radio-k-number of G is the smallest integer s for which there exists a function f:V(G)→{0,1,2,…,s} such that for any two vertices u,v∈V(G), |f(u)−f(v)|⩾k+1−d(u,v), where d(u,v) is the distance between u and v. In particular, when d is the diameter of G, the radio-d-number is called the radio number of G. This article contains four major parts.

An on-line chain partitioning algorithm receives the elements of a poset one at a time, and when an element is received, irrevocably assigns it to one of the chains. In this paper, we present an on-line algorithm that partitions posets of width w into wO(loglogw) chains. This improves over previously best known algorithms using wO(logw) chains by Bosek and Krawczyk and by Bosek, Kierstead, Krawczyk

A (finite) graph is odd if all its vertices have odd degrees. The principal aim of this survey is to present the current state of research on covers and decompositions of graphs into fewest possible number of odd subgraphs. Given a graph G, the parameters χo′(G) and covo(G) denote, respectively, the minimum size of a decomposition and cover of G consisting of odd subgraphs. Pyber (1991) and Mátrai

A perfect code in a graph Γ=(V,E) is a subset C of V such that no two vertices in C are adjacent and every vertex in V∖C is adjacent to exactly one vertex in C. A subgroup H of a group G is called a subgroup perfect code of G if there exists a Cayley graph of G which admits H as a perfect code. Equivalently, H is a subgroup perfect code of G if there exists an inverse-closed subset A of G containing

Let G be a graph, and let w be a positive real-valued weight function on V(G). For every subset S of V(G), let w(S)=∑v∈Sw(v). A non-empty subset S⊂V(G) is a weighted safe set of (G,w) if, for every component C of the subgraph induced by S and every component D of G−S, we have w(C)≥w(D) whenever there is an edge between C and D. If the subgraph of G induced by a weighted safe set S is connected, then

Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erdős–Gallai Theorem in random graphs. In particular, we determine, up to a constant factor, the maximum number of edges in a Pn-free subgraph of G(N,p), practically for all values of N,n and p. Our work is also motivated by the recent progress on the size-Ramsey

Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths – a result that shows a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views on classes with bounded linear rankwidth and prove structural

This paper surveys recent development of concepts related to coloring of signed graphs. Various approaches are presented and discussed.

Introduced by Dvořák and Postle, the notion of DP-coloring generalizes list coloring and helps to prove new results on list coloring. We consider 1-defective and 2-defective DP-colorings of graphs with 2 colors. For j=1,2, we find exact lower bounds on the number of edges in (j,2)-DP-critical graphs (that is, graphs that do not admit j-defective DP-colorings with 2 colors but whose every proper subgraph

The generalized coloring numbers colr(G) (also denoted by scolr(G)) and wcolr(G) of a graph G were introduced by Kierstead and Yang as a generalization of the usual coloring number, and have found important theoretical and algorithmic applications. For each distance r, these numbers are determined by an “optimal” ordering of the vertices of G. We study the question of whether it is possible to find

We extend Edmonds’ Branching Theorem to locally finite infinite digraphs. As examples of Oxley or Aharoni and Thomassen show, this cannot be done using ordinary arborescences, whose underlying graphs are trees. Instead we introduce the notion of pseudo-arborescences and prove a corresponding packing result. Finally, we verify some tree-like properties for these objects, but give also an example that

This survey presents some combinatorial problems with number-theoretic flavor. Our journey starts from a simple graph coloring question, but at some point gets close to dangerous territory of the Riemann Hypothesis. We will mostly focus on open problems, but we will also provide some simple proofs, just for adorning.

Edmonds’ fundamental theorem on arborescences characterizes the existence of k pairwise arc-disjoint spanning arborescences with prescribed root sets in a digraph. In this paper, we study the problem of packing branchings in digraphs under cardinality constraints on their root sets by arborescence augmentation. Let D=(V+x,A) be a digraph, P= {I1,…,Il} be a partition of [k], c1,…,cl,c1′,…,cl′ be nonnegative

For a pair of natural numbers k,l, a (k,l)-colouring of a graph G is a partition of the vertex set of G into (possibly empty) sets S1,S2,…,Sk, C1,C2,…,Cl such that each set Si is an independent set and each set Cj induces a clique in G. The (k,l)-colouring problem, which is NP-complete in general, has been studied for special graph classes such as chordal graphs, cographs and line graphs. Let κˆ(G)=(κ0(G)

Assume G is a graph and S is a set of permutations of integers. An S-labelling of G is a pair (D,σ), where D is an orientation of G and σ:E(D)→S is a mapping which assigns to each arc e of D a permutation σe∈S. A proper k-colouring of (D,σ) is a mapping f:V(G)→[k]={1,2,…,k} such that σe(f(x))≠f(y) for each arc e=(x,y). We say G is S-k-colourable if any S-labelling (D,σ) of G has a proper k-colouring

For a graph H, a graph G is H-induced-saturated if G does not contain an induced copy of H, but either removing an arbitrary edge from G or adding an arbitrary non-edge to G creates an induced copy of H. Depending on the graph H, an H-induced-saturated graph does not necessarily exist. In fact, (Martin and Smith, 2012) showed that P4-induced-saturated graphs do not exist, where Pk denotes a path on

Zero forcing is a deterministic iterative graph coloring process in which vertices are colored either blue or white, and in every round, any blue vertices that have a single white neighbor force these white vertices to become blue. Here we study probabilistic zero forcing, where blue vertices have a non-zero probability of forcing each white neighbor to become blue. We explore the propagation time

Barnette identified two interesting classes of cubic polyhedral graphs for which he conjectured the existence of a Hamiltonian cycle. We examine these classes from the point of view of distance-two colourings. A distance-two r-colouring of a graph G is an assignment of r colours to the vertices of G so that any two vertices at distance at most two have different colours. A cubic graph obviously needs

We give an upper bound on the number of cycles in a simple graph in terms of its degree sequence, and apply this bound to resolve several conjectures of Király (2009) and Arman and Tsaturian (2017) and to improve upper bounds on the maximum number of cycles in a planar graph.

Consider all k-element subsets and ℓ-element subsets (k>ℓ) of an n-element set as vertices of a bipartite graph. Two vertices are adjacent if the corresponding ℓ-element set is a subset of the corresponding k-element set. Let Gk,ℓ denote this graph. The domination number of Gk,1 was exactly determined by Badakhshian, Katona and Tuza. A conjecture was also stated there on the asymptotic value (n tending

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