For an infinite sequence A of positive integers and a configuration m={(k1,m1),⋯,(kl,ml)} with degree s>1, let Rm(A,n) be the number of different solutions of the equation k1(a1,1+⋯+a1,m1)+⋯+kl(al,1+⋯+al,ml)≤n with ai,j∈A. Let ε>0. Rué proved that Rm(A,n)=cn+O(n1∕4−ε) cannot hold for any constant c>0. In this paper, we improve this result.

The notion of symmetric and asymmetric peaks in Dyck paths was introduced by Flórez and Rodríguez, who counted the total number of such peaks over all Dyck paths of a given length. In this paper we generalize their results by giving multivariate generating functions that keep track of the number of symmetric peaks and the number of asymmetric peaks, as well as the widths of these peaks. We recover

Consider a quadratic cone K in the 3-dimensional projective space PG(3,q) over a finite field of order q, where q is a prime power. Let E (respectively, T, S) denote the set of all lines of PG(3,q) that are external (respectively, tangent, secant) with respect to K. We characterize the minimum size blocking sets in PG(3,q) with respect to the line set A, where A is one of E, T, S, E∪T, E∪S and T∪S

In 1965, Vizing proved that every planar graph G with maximum degree Δ≥8 is edge Δ-colorable. It is also proved that every planar graph G with maximum degree Δ=7 is edge Δ-colorable by Sanders and Zhao, independently by Zhang. In this paper, we extend the above results by showing that every K5-minor free graph with maximum degree Δ at least seven is edge Δ-colorable.

A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of k-level vertices in an arrangement of n hyperplanes in Rd (vertices with exactly k of the hyperplanes passing below them). This is essentially a dual version of the k-set problem, which, in a primal setting, seeks bounds for the maximum number of k-sets determined by n points in Rd, where

Chordal graphs are important in the structural and algorithmic graph theory. A digraph analogue of chordal graphs was introduced by Haskin and Rose in 1973 but has not been the subject of active studies until recently when a characterization of semicomplete chordal digraphs in terms of forbidden subdigraphs was found by Meister and Telle. Locally semicomplete digraphs, quasi-transitive digraphs, and

Let vi be vectors in Rd and {εi} be independent Rademacher random variables. Then the Littlewood–Offord problem entails finding the best upper bound for supx∈RdP(∑εivi=x). Generalizing the uniform bounds of Littlewood–Offord, Erdős and Kleitman, a recent result of Dzindzalieta and Juškevičius provides a non-uniform bound that is optimal in its dependence on ‖x‖2. In this short note, we provide a simple

An equitable k-partition of a graph G is a collection of induced subgraphs (G[V1],G[V2],…,G[Vk]) of G such that (V1,V2,…,Vk) is a partition of V(G) and −1≤|Vi|−|Vj|≤1 for all 1≤i

The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with possibly different maximum fields of linearity. We also consider the intersection between a certain linear set of maximum rank and any other linear set of the same rank. The strategy relies on the study

For an integer i≥1, Zi is the graph obtained by attaching an endvertex of a path of length i to a vertex of a triangle. We prove that every 3-connected {K1,3,Z7}-free graph is Hamilton-connected, with one exceptional graph. The result is sharp.

Let G be a simple undirected graph. The permanental sum of G is equal to the permanent of the matrix I+A(G), where I is the identity matrix and A(G) is an adjacency matrix of G. The computation of permanental sum is #P-complete. In this paper, we compute the permanental sum of graphs of small sizes and derive explicit formulae for the permanental sum of graphs of large sizes. The results from small

We make some progresses on Saxl Conjecture. Firstly, we show that the probability that a partition is comparable in dominance order to the staircase partition tends to zero as the staircase partition grows. Secondly, for partitions whose Durfee size is k where k≥3, by semigroup property, we show that there exists a number nk such that if the tensor squares of the first nk staircase partitions contain

A well-known technique to construct regular graphs with girth 5 is the amalgamation into the incidence graphs Cq and Lq, elliptic semiplanes of type C and L respectively, where q is a prime power. The case q odd has extensively been studied by means of amalgamations into Lq. In this paper we provide new families of small regular graphs of girth 5 constructed by amalgamation into Cq using finite fields

Let F2m be the finite field of 2m elements and s be any positive integer. The existing literature only gives an effective calculation method to represent all distinct Euclidean self-dual cyclic codes of length 2s over the finite chain ring F2m+uF2m (u2=0), such as in Cao et al., (2019). As a development of this topic, we provide an explicit expression for each of these self-dual cyclic codes, using

The γ-coefficients of Eulerian polynomials were first considered by Foata and Schützenberger. In this paper, we provide combinatorial interpretations for the γ-coefficients arising from the segmented permutations and segmented derangements via Brändén’s modified Foata–Strehl action. We also give the combinatorial interpretations of γ-coefficients for the (>,≤,−)-avoiding inversion sequences via continued

A uniformly-ordered ρ-labeling (also known as a ρ++-labeling) of a bipartite graph was introduced by El-Zanati, Vanden Eynden, and Punnim. Such a labeling of a bipartite graph G with m edges yields a cyclic G-decomposition of K2mt+1 for every positive integer t. Here we show that for every even integer n≥8, the generalized Petersen graph P(n,3) admits a ρ++-labeling and hence cyclically decomposes

A wheel graph is a graph formed by connecting a single vertex to all vertices of a cycle. A graph is called wheel-free if it does not contain any wheel graph as a subgraph. In 2010, Nikiforov proposed a Brualdi–Solheid–Turán type problem: what is the maximum spectral radius of a graph of order n that does not contain subgraphs of particular kind. In this paper, we study the Brualdi–Solheid–Turán type

A graph is k-planar if it can be drawn in the plane so that each edge is crossed at most k times. Typically, the class of 1-planar graphs is among the most investigated graph families within the so-called “beyond planar graphs”. A dynamic ℓ-list coloring of a graph is a proper coloring so that each vertex receives a color from a list of ℓ distinct candidate colors assigned to it, and meanwhile, there

We show that any loopy multigraph with a graphical degree sequence can be transformed into a simple graph by a finite sequence of double edge swaps with each swap involving at least one loop or multiple edge. Our result answers a question of Janson motivated by random graph theory, and it adds to the rich literature on reachability of double edge swaps with applications in Markov chain Monte Carlo

A graph is strongly perfect if every induced subgraph H has a stable set that meets every nonempty maximal clique of H. The characterization of strongly perfect graphs by a set of forbidden induced subgraphs is not known. Here we provide several new minimal non-strongly-perfect graphs.

Let f(n,H) denote the maximum number of copies of H in an n-vertex planar graph. The order of magnitude of f(n,Pk), where Pk is a path on k vertices, is n⌊k−12⌋+1. In this paper we determine the asymptotic value of f(n,P5) and give conjectures for longer paths.

We provide new values of the bipartite Ramsey number RB(C4,K1,n) using induced subgraphs of the incidence graph of a projective plane. The approach, based on deleting subplanes of projective planes, has been used in related extremal problems and allows us to unify previous results and extend them. More importantly, using deep stability results on 2modp sets and double blocking sets, we can show some

Felsner, Li and Trotter showed that the dimension of the adjacency poset of an outerplanar graph is at most 5, and gave an example of an outerplanar graph whose adjacency poset has dimension 4. We improve their upper bound to 4, which is then best possible.

Let G be a t-uniform hypergraph, and let c(G) denote the cyclic index of the adjacency tensor of G. Let m,s be positive integers such that s≥2 and m=st. The generalized power Gm,s of G is obtained from G by blowing up each vertex into an s-set and preserving the adjacency relation. It was conjectured that c(Gm,s)=s⋅c(G). In this paper, by using a matrix equation over Zm that characterizes the spectral

Let G be a non-trivial simple graph with vertices V(G)=V and edges E(G)=E, and let n=|V|,m=|E|. Computing the diameter of G and the min–max center of G (C(G)) are both quadratic-time (O(m2)). A problem is strongly subquadratic-time if it is O(m2−ϵ) for some ϵ>0. If either the diameter problem or the center problem is strongly subquadratic-time, then the Strong Exponential Time Hypothesis would be violated

The f-vector of a polytope consists of the numbers of its i-dimensional faces. An open field of study is the characterization of all possible f-vectors. It has been solved in three dimensions by Steinitz in the early 19th century. We state a related question, i.e., to characterize f-vectors of three dimensional polytopes respecting a symmetry, given by a finite group of matrices. We give a full answer

Let (G,+) be an abelian group. A subset of G is sumfree if it contains no elements x,y,z such that x+y=z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The classical inequality S(n)≤Rn(3)−2, between the Schur number S(n) and the Ramsey number Rn(3)=R(3,…,3), is shown to remain valid in a wider context, involving the Schur degree

We say G→(C,Pn) if G−E(F) contains an n-vertex path Pn for any spanning forest F⊂G. The size Ramsey number Rˆ(C,Pn) is the smallest integer m such that there exists a graph G with m edges for which G→(C,Pn). Dudek, Khoeini and Prałat proved that for sufficiently large n, 2.0036n≤Rˆ(C,Pn)≤31n. In this note, we improve both the lower and upper bounds to 2.066n≤Rˆ(C,Pn)≤5.25n+O(1). Our construction for

By the theorem of Mantel (1907) it is known that a graph with n vertices and ⌊n24⌋+1 edges must contain a triangle. A theorem of Erdős gives a strengthening: there are not only one, but at least ⌊n2⌋ triangles. We give a further improvement: if there is no vertex contained by all triangles then there are at least n−2 of them. There are some natural generalizations when (a) complete graphs are considered

In this paper we show that the rank of every chiral polytope having a Suzuki group as automorphism group is 3. This gives a positive answer to a conjecture of Isabel Hubard and Dimitri Leemans.

We consider q-ary (linear and nonlinear) block codes with exactly two distances: d and d+δ. We derive necessary conditions for existence of such codes (similar to the known conditions in the projective case). In the linear (but not necessary projective) case, we prove that under certain conditions the existence of such linear 2-weight code with δ>1 implies the following equality of greatest common

We show that the Waring number over a finite field Fq, denoted as g(k,q), when exists coincides with the diameter of the generalized Paley graph Γ(k,q)=Cay(Fq,Rk) with Rk={xk:x∈Fq∗}. We find infinite new families of exact values of g(k,q) from a characterization of graphs Γ(k,q) which are also Hamming graphs proved by Lim and Praeger in 2009. Then, we show that every positive integer is the Waring

For a digraph G and v∈V(G), let δ+(v) be the number of out-neighbors of v in G. The Caccetta–Häggkvist conjecture states that for all k≥1, if G is a digraph with n=|V(G)| such that δ+(v)≥k for all v∈V(G), then G contains a directed cycle of length at most ⌈n∕k⌉. In Aharoni et al. (2019), Aharoni proposes a generalization of this conjecture, that a simple edge-colored graph on n vertices with n color

Let n>s>0 be integers, X an n-element set and A,ℬ⊂2X two families. If |A∪B|≤s for all A∈A,B∈ℬ, then A and ℬ are called cross s-union. Assuming that neither A nor ℬ is empty, we prove several best possible bounds. In particular, we show that |A|+|ℬ|≤1+∑0≤i≤sni. Supposing n≥2s and A,ℬ are antichains, we show that |A|+|ℬ|≤n1+ns−1 unless A={0̸} or ℬ={0̸}. An analogous result for three families is established

A finite simple graph G is called a sum graph (integral sum graph) if there is a bijection f from the vertices of G to a set of positive integers S (a set of integers S) such that uv is an edge of G if and only if f(u)+f(v)∈S. For a connected graph G, the sum number (the integral sum number) of G, denoted by σ(G) (ζ(G)), is the minimum number of isolated vertices that must be added to G so that the

In this paper, we characterize 2-connected {K1,3,N3,1,1}-free graphs without Hamiltonian cycle, where K1,3 is the star of order 4 and Nn1,n2,n3 is the graph obtained from K3 and three vertex-disjoint paths Pn1+1, Pn2+1, Pn3+1 by identifying each of vertices of K3 with an endvertex of one of the paths. Such a characterization gives some refinements for known results, for example, a characterization

We introduce in this paper leprechauns, fairy chess pieces that can move either like the standard queen, or to any tile within k king moves. We then study the problem of placing n leprechauns on an n×n chessboard. When k=1, this is equivalent to the standard n-Queens Problem. We solve the problem for k=2, as well as for k>2 and n≤(k+1)2, giving in the process an upper bound on the lowest non-trivial

Let R=F2+uF2 (u2=0) and S=F2+uF2+u2F2 (u3=0) be two finite commutative chain rings. This paper studies F2RS-cyclic codes, which are described as S[x]-submodules of the S[x]-module F2[x]∕〈xr−1〉×R[x]∕〈xs−1〉×S[x]∕〈xt−1〉. We study their generator polynomials and the minimal generating sets. We classify each case of the generating sets separately and determine the size of each such case. Free F2RS-cyclic

Random spanning trees of a graph G are governed by a corresponding probability mass distribution (or “law”), μ, defined on the set of all spanning trees of G. This paper addresses the problem of choosing μ in order to utilize the edges as “fairly” as possible. This turns out to be equivalent to minimizing, with respect to μ, the expected overlap of two independent random spanning trees sampled with

We study properties of the volume of projections of the n-dimensional cross-polytope ♢n={x∈Rn∣|x1|+⋯+|xn|⩽1}. We prove that the projection of ♢n onto a k-dimensional coordinate subspace has the maximum possible volume for k=2 and for k=3. We obtain the exact lower bound on the volume of such a projection onto a two-dimensional plane. Also, we show that there exist local maxima which are not global

We define and study odd analogues of classical geometric and combinatorial objects associated to permutations, namely odd Schubert varieties, odd diagrams, and odd inversion sets. We show that there is a bijection between odd inversion sets of permutations and acyclic orientations of the Turán graph, that the dimension of the odd Schubert variety associated to a permutation is the odd length of the

As a generalization of list coloring, DP-coloring of graphs was introduced by Dvořák and Postle (2018). Recently, Bernshteyn and Kostochka introduced edge DP-coloring of graphs which is naturally corresponding to the DP-coloring of their line graphs. Let χDP′(G) denote the edge DP-chromatic number of a graph G. In this paper, we prove that if G is a planar graph with maximum degree Δ and without cycles

Let G be an infinite, vertex-transitive lattice with degree λ and fix a vertex on it. Consider all cycles of length exactly l from this vertex to itself on G. Erasing loops chronologically from these cycles, what is the fraction Fp∕λℓ(p) of cycles of length l whose last erased loop is some chosen self-avoiding polygon p of length ℓ(p), when l→∞ ? We use combinatorial sieves to prove an exact formula

In this work we study the operations of handle slides introduced recently for delta-matroids by Iain Moffatt and Eunice Mphako-Banda. We then prove that any class of delta-matroids that is closed under handle slides is a subclass of the class of binary delta-matroids.

Let G be a graph on n vertices and λ1,λ2,…,λn its eigenvalues. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. In this paper, we present some new lower bounds for the Estrada index of graphs and in particular of bipartite graphs that only depend on the number of vertices, the number of edges, Randić index, maximum and minimum degree

The Ramsey number RX(p,q) for a class of graphs X is the minimum n such that every graph in X with at least n vertices has either a clique of size p or an independent set of size q. We say that Ramsey numbers are linear in X if there is a constant k such that RX(p,q)≤k(p+q) for all p,q. In the present paper we conjecture that if X is a hereditary class defined by finitely many forbidden induced subgraphs

Given two graphs G and H, the rainbow number rb(G,H) for H with respect to G is defined as the minimum number k such that any k-edge-coloring of G contains a rainbow H, i.e., a copy of H, all of its edges have different colors. Denote by Mt a matching of size t and Tn the class of all plane triangulations of order n, respectively. Jendrol′ et al. initiated to investigate the rainbow numbers for matchings

In 1980, Albertson and Berman introduced partial coloring. In 2000, Albertson, Grossman, and Haas introduced partial list coloring. Here, we initiate the study of partial coloring for an insightful generalization of list coloring introduced in 2015 by Dvořák and Postle, DP-coloring (or correspondence coloring). We consider the DP-coloring analogue of the Partial List Coloring Conjecture, which generalizes

Given a 3-colorable graph X, the 3-coloring complex B(X) is the graph whose vertices are all the independent sets which occur as color classes in some 3-coloring of X. Two color classes C,D∈V(B(X)) are joined by an edge if C and D appear together in a 3-coloring of X. The graph B(X) is 3-colorable. Graphs for which B(B(X)) is isomorphic to X are termed reflexive graphs. In this paper, we consider 3-edge-colorings

In 2002, Björner and de Longueville showed the neighborhood complex of the 2-stable Kneser graph KG(n,k)2−stab has the same homotopy type as the (n−2k)-sphere. A short time ago, an analogous result about the homotopy type of the neighborhood complex of almost s-stable Kneser graph has been announced by the second author. Combining this result with the famous Lovász’s topological lower bound on the

A generalised quadrangle is a point–line incidence geometry G such that: (i) any two points lie on at most one line, and (ii) given a line L and a point p not incident with L, there is a unique point on L collinear with p. They are a specific case of the generalised polygons introduced by Tits (1959), and these structures and their automorphism groups are of some importance in finite geometry. An integral

Magic square is an ancient and important subject in combinatorial mathematics, and many kinds of magic square are studied and concerned by many scholars, in particular, the existence of normal bimagic squares has been studied for over one hundred years. In this paper, the existence of the normal bimagic squares of even order is investigated, the ideas of general row (column) bimagic rectangles and

Tuza (1981) conjectured that the size τ(G) of a minimum set of edges that intersects every triangle of a graph G is at most twice the size ν(G) of a maximum set of edge-disjoint triangles of G. In this paper we present three results regarding Tuza’s Conjecture. We verify it for graphs with treewidth at most 6; we show that τ(G)≤32ν(G) for every planar triangulation G different from K4; and that τ(G)≤95ν(G)+15

A permutation class C is said to be splittable if there exist two proper subclasses A,B⊊C such that any σ∈C can be red–blue colored so that the red (respectively, blue) subsequence of σ is order isomorphic to an element of A (respectively, B). The class C is said to be composable if there exists some number of proper subclasses A1,…,Ak⊊C such that any σ∈C can be written as α1∘⋯∘αk for some αi∈Ai. We

For positive integers s,t,u,v, we define a bipartite graph ΓR(XsYt,XuYv) where each partite set is a copy of R3, and a vertex (a1,a2,a3) in the first partite set is adjacent to a vertex [x1,x2,x3] in the second partite set if and only if a2+x2=a1sx1tanda3+x3=a1ux1v.In this paper, we classify all such graphs according to girth.

Simultaneous core partitions have been extensively exploited after Anderson’s work on the enumeration of (s,t)-core partitions. Ford, Mai and Sze established a bijection between self-conjugate (s,t)-core partitions and lattice paths in the ⌊s2⌋×⌊t2⌋ rectangle consisting of north and east steps, thereby showing that the number of such partitions is given by ⌊s2⌋+⌊t2⌋⌊s2⌋ for relatively prime integers

A d-dimensional brick is a set I1×⋯×Id where each Ii is an interval. Given a brick B, a brick partition of B is a partition of B into bricks. A brick partition Pd of a d-dimensional brick is k-piercing if every axis–parallel line intersects at least k bricks in Pd. Bucic et al. (2019) explicitly asked the minimum size p(d,k) of a k-piercing brick partition of a d-dimensional brick. The answer is known

We give an elementary proof of an interesting combinatorial identity which is of particular interest in graph theory and its applications. Two applications to enumeration of forests with closed-form expressions are given.

A triple intersection number in a finite generalized hexagon S of order (s,t) is a number of the form |Γi(x)∩Γj(y)∩Γk(ω)|, where x, y are two points and ω is either a point z or a line L. The fact that S is extremal, i.e. satisfies t=s3>1, implies that certain combinatorial properties regarding triple intersection numbers hold. The earliest result in this direction is due to Haemers who showed that

Assume that G is a finite group and let a and b be non-negative integers. We define an undirected graph Γa,b(G) whose vertices correspond to the elements of Ga∪Gb and in which two tuples (x1,…,xa) and (y1,…,yb) are adjacent if and only if 〈x1,…,xa,y1,…,yb〉=G. Our aim is to estimate the genus, the thickness and the crossing number of the graph Γa,b(G) when a and b are positive integers, giving explicit