Let K be a set of positive integers and let G be an additive group. A (G,K,1) difference packing is a set of subsets of G with sizes from K whose list of differences covers every element of G at most once. It is balanced if the number of blocks of size k∈K does not depend on k. In this paper, we determine a balanced (Z4u×Z8v,{4,5},1) difference packing of the largest possible size whenever uv is odd

Let m be a positive integer, and γ be a non-zero element of F2m. The Lee distance of all cyclic codes and (1+uγ)-constacyclic codes of length 2s over F2m+uF2m is completely determined. In particular, it is shown that the Lee distance of such codes is independent of the choice of a trace orthogonal basis of F2m.

DP-coloring was introduced by Dvořák and Postle as a generalization of list coloring. It was originally used to solve a longstanding conjecture by Borodin, stating that every planar graph without cycles of lengths 4 to 8 is 3-choosable. In this paper, we give three sufficient conditions for a planar graph to be DP-4-colorable. Actually all the results (Theorem 1.3, Theorem 1.4, Theorem 1.7) are stated

Given a family of graphs F, a graph G is said to be F-saturated if G does not contain a copy of F as a subgraph for any F∈F, but the addition of any edge e∉E(G) creates at least one copy of some F∈F within G. The minimum size of an F-saturated graph on n vertices is called the saturation number, denoted by sat(n,F). Let C≥r be the family of cycles of length at least r. Ferrara et al. (2012) gave lower

In this paper, we consider the set of all domino tilings of a cubiculated region. The primary question we explore is: How can we move from one tiling to another? Tiling spaces can be viewed as spaces of subgraphs of a fixed graph with a fixed degree sequence. Moves to connect such spaces have been explored in algebraic statistics. Thus, we approach this question from an applied algebra viewpoint, making

In this paper we give two characterizations of the p×q-grid graphs as co-edge-regular graphs with four distinct eigenvalues.

Given a bipartite graph with bipartition (A,B) where B is equipartitioned into k blocks, can the vertices in A be picked one by one so that at every step, the picked vertices cover roughly the same number of vertices in each of these blocks? We show that, if each block has cardinality m, the vertices in B have the same degree, and each vertex in A has at most cm neighbors in every block where c>0 is

A sequence (xn) on the unit interval is said to have Poissonian pair correlation if #{1≤i≠j≤N:‖xi−xj‖≤s/N}=2sN(1+o(1)) for all reals s>0, as N→∞. It is known that, if (xn) has Poissonian pair correlations, then the number g(n) of different gap lengths between neighboring elements of {x1,…,xn} cannot be bounded along any index subsequence (nt). First, we improve this by showing that, if (xn) has Poissonian

A Gallai-coloring (Gallai-k-coloring) is an edge-coloring (with colors from {1,2,…,k}) of a complete graph without rainbow triangles. Given a graph H and a positive integer k, the k-colored Gallai-Ramsey number GRk(H) is the minimum integer n such that every Gallai-k-coloring of the complete graph Kn contains a monochromatic copy of H. In this paper, we consider two extremal problems related to Gallai-k-colorings

Andrews, Lewis and Lovejoy introduced the partition function PD(n) as the number of partitions of n with designated summands. In a recent work, Lin studied a partition function PDt(n) which counts the number of tagged parts over all the partitions of n with designated summands. He proved that PDt(3n+2) is divisible by 3. In this paper, we first introduce a structure named partitions with overline designated

We construct an evidently positive multiple series as a generating function for partitions satisfying the multiplicity condition in Schur's partition theorem. Refinements of the series when parts in the said partitions are classified according to their parities or values mod 3 are also considered. Direct combinatorial interpretations of the series are provided.

We focus on writing double sum representations of the generating functions for the number of partitions satisfying some gap conditions. Some example sets of partitions to be considered are partitions into distinct parts and partitions that satisfy the gap conditions of the Rogers–Ramanujan, Göllnitz–Gordon, and little Göllnitz theorems. We refine our representations by imposing a bound on the largest

Let H(n,q2) be a non–degenerate Hermitian variety of PG(n,q2), n≥2. Let NU(n+1,q2) be the graph whose vertices are the points of PG(n,q2)∖H(n,q2) and two vertices P1,P2 are adjacent if the line joining P1 and P2 is tangent to H(n,q2). Then NU(n+1,q2) is a strongly regular graph. In this paper we show that NU(n+1,q2), n≠3, is not determined by its spectrum.

In n-dimensional affine space over the field Z/3Z, a cap is given by a set of points no three of which are in a line, and the cap set problem asks for the largest possible size of an arbitrary cap. The solution to the cap set problem is known for n at most 6. In this paper, we define and apply standard diagrams. These pictures interpret a well-known technique for solving the cap set problem in a new

A subgraph of an edge-colored graph is rainbow, if all of its edges have different colors. For a graph G and a family H of graphs, the anti-Ramsey number ar(G,H) is the maximum number k such that there exists an edge-coloring of G with exactly k colors without rainbow copy of any graph in H. In this paper, we study the anti-Ramsey numbers of C3 and C4 in complete r-partite graphs. For r≥3 and n1≥n2≥⋯≥nr≥1

A graph is symmetric if its automorphism group acts transitively on both the vertex and the arc sets. Stimulated by Lorimer's work on symmetric graphs of prime valency, we make a further investigation on the structural properties of the automorphism groups of connected symmetric graphs with prime valency. Also, as an application, we give a classification for symmetric graphs of prime valency arising

Let Γ=(V,E) be a graph of order n. A distance magic labeling of Γ is a bijection ℓ:V→{1,2,…,n} for which there exists a positive integer k such that ∑x∈N(u)ℓ(x)=k for all vertices u∈V, where N(u) is the neighborhood of u. A graph is said to be distance magic if it admits a distance magic labeling. In this paper we classify all connected tetravalent distance magic circulants, that is Cayley graphs Cay(Zn;S)

In this paper, we prove that for a sufficiently large integer d and a connected graph G, if |V(G)|<(d+2)(δ(G)+1)3, then there exists a spanning tree T of G such that diam(T)≤d.

The paper is devoted to the problem of enumerating r-regular maps on two simplest non-orientable surfaces, the projective plane and the Klein bottle, up to all symmetries (so-called unsensed maps). We obtain general formulas that reduce the problem of counting such maps to the problem of enumerating rooted quotient maps on orbifolds. In addition, we solve the problem of explicitly describing all cyclic

The Turán number T(n,α+1,r) is the minimum number of edges in an n-vertex r-graph whose independence number does not exceed α. For each r≥2, there exists t⁎(r) such that T(n,α+1,r)=t⁎(r)nrα1−r(1+o(1)) as α/r→∞ and n/α→∞. It is known that t⁎(2)=1/2, and the conjectured value of t⁎(3) is 2/3. We prove that t⁎(4)<0.706335.

Let n≥2 be an integer. The graph G(n) is obtained by letting all the elements of {0,…,n−1} to be the vertices and defining distinct vertices x and y to be adjacent if and only if gcd⁡(x+y,n)=1. In this paper, well-coveredness, Cohen–Macaulayness, vertex-decomposability and Gorensteinness of these graphs and their complements are characterized. These characterizations provide large classes of Cohen–Macaulay

The L-index (resp. Q-index) of a graph G is the largest eigenvalue of the Laplacian matrix (resp. signless Laplacian matrix) of G. In this paper, we determine the maximal Q-index (and L-index) of connected graphs with fixed size and diameter. As by-products, we deduce a result due to Lal and Patra (2007) [9] regarding the maximal L-index of trees with fixed order (or size) and diameter.

In 1984, Matthews and Sumner conjectured that every 4-connected, claw-free graph contains a Hamiltonian cycle. This still unresolved conjecture has been the motivation for research into the existence of other cycle structures. In this paper, we consider the stronger property of pancyclicity for 4-connected graphs. In particular, we show that every 4-connected, {K1,3,N(i,j,k)}-free graph, where i,j

This paper investigates a topic inspired by a magic trick called the “Tantalizer”, which is a card game resembling the well-known Josephus Problem. We study the spectator-first Tantalizer problem with a deck of n-cards and investigate which card is left in the end after a series of dealing operations. A formula and an algorithm with a running time complexity O(log⁡n) based on the binary form of n are

The paper studies flag-transitive groups of automorphisms of 2-(v,k,λ) designs D with (r−λ,k)=1. In 1968, Peter Dembowski proved that such groups of automorphisms act primitively on the set of points of D. The purpose of the present paper is to reduce the structure of these groups even more, we show that they are affine or almost simple.

In this paper, we present two explicit expressions of the moments M2k of the Kesten distribution in terms of Catalan's triangles Tk,i=k−i+1k+1(k+ik) and Shapiro's Catalan triangles Bk,j=jk(2kk−j), respectively. As a corollary, we provide a generalization of the identity, due to Eplett, between the Catalan number Ck=1k+1(2kk) and the triangles Bk,j. As an application, we obtain an asymptotic formula

The topic is the average order A(G) of a connected induced subgraph of a graph G. This generalizes, to graphs in general, the average order of a subtree of a tree. In 1983 Jamison proved that the average order, over all trees of order n, is minimized by the path Pn, the average being A(Pn)=(n+2)/3. In 2018, Kroeker, Mol, and Oellermann conjectured that Pn minimizes the average order over all connected

We say that G has the property P if G−N[v] is a cycle for any vertex v∈V(G), where N[v] is the closed neighborhood of v in G. For an integer l∈{3,4,5,6}, let Gl be a set of graphs defined as follows: G3={2K3}∪{G:both G and G‾ are connected, and G‾ is a triangle-free cubic graph}, where H‾ denotes the complement of H, G4={L(H)‾:H is a connected triangle-free cubic graph}, where L(H) denotes the line

Let G be a finite undirected simple connected graph with vertex set V(G), distance function ∂ and diameter d. Let D⊆{0,1,…,d} be a set of distances. A bijection l:V(G)→{1,2,…,|V(G)|} is called a D-magic labeling of G if there exists a constant k such that ∑x∈ND(v)l(x)=k for any vertex v∈V(G), where ND(v)={x∈V(G):∂(x,v)∈D}. We say G has a D-magic labeling if G admits a D-magic labeling. In this paper

A graph drawn on the plane is 2-immersed in the plane if each edge is crossed by at most two other edges. By a proper 2-immersion of a graph we mean a 2-immersion of the graph in the plane such that there is at least one crossing point. We consider the class T of all finite graphs triangulating the plane such that the graphs have no loops and multiple edges, the vertices have degree 5 and 6 only, and

We construct qubit stabilizer codes with parameters 〚81,0,20〛2 and 〚94,0,22〛2 for the first time. We use symplectic self-dual additive codes over F4 built by modifying the adjacency matrices of suitable metacirculant graphs found by a randomized search procedure.

Minimal codewords have applications in decoding linear codes and in cryptography. We study the maximum number of minimal codewords in binary linear codes of a given length and dimension. Improved lower and upper bounds on the maximum number are presented. We determine the exact values for the case of linear codes of dimension k and length k+2 and for small values of the length and dimension. We also

Let G be a finite, connected graph. The average distance of a vertex v of G is the arithmetic mean of the distances from v to all other vertices of G. The remoteness ρ(G) and the proximity π(G) of G are the maximum and the minimum of the average distances of the vertices of G. In this paper, we present a sharp upper bound on the remoteness of a triangle-free graph of given order and minimum degree

Let E be a possibly infinite set and let M and N be matroids defined on E. We say that the pair {M,N} has the Intersection property if M and N share an independent set I admitting a bipartition IM⊔IN such that spanM(IM)∪spanN(IN)=E. The Matroid Intersection Conjecture of Nash-Williams says that every matroid pair has the Intersection property. The conjecture is known and easy to prove in the case when

For a graph G and a family of graphs H, the Turán number ex(G,H) is defined to be the maximum number of edges among all H-free subgraphs of G. Inverting this problem, Briggs and Cox (2019) [5] studied the extremal function εH(k)=sup{e(G)|ex(G,H)

A graph is 1-embeddable on a closed surface if there exists a drawing of the graph on the surface such that each edge crosses at most one other edge at a point. In this paper, we determine all the 1-embeddable complete k-partite graphs on the projective plane.

The existence of a biplane with parameters (121,16,2) is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of a possible biplane D of order 14 divides 27⋅32⋅5⋅7⋅11⋅13. In this paper we show that such a biplane does not have an automorphism of order 11 or 13, and thereby establish that |Aut(D)| divides 27⋅32⋅5⋅7. Further, we study

A graph G is said to be 2-distinguishable if there is a labeling of the vertices with two labels so that only the trivial automorphism preserves the labels. The minimum size of a label class, over all 2-distinguishing labelings, is called the cost of 2-distinguishing, denoted by ρ(G). For n≥4 the hypercubes Qn are 2-distinguishable, but the values for ρ(Qn) have been elusive, with only bounds and partial

Our main goal is to develop a representation for finite distributive nearlattices through certain ordered structures. This representation generalizes the well-known representation given by Birkhoff for finite distributive lattices through finite posets. We also study finite distributive nearlattices through the concepts of dual atoms, boolean elements, complemented elements and irreducible elements

For a graph H and a k-chromatic graph F, if the Turán graph Tk−1(n) has the maximum number of copies of H among all n-vertex F-free graphs (for n large enough), then H is called F-Turán-good, or k-Turán-good for short if F is Kk. In this paper, we construct some new classes of k-Turán-good graphs and prove that P4 and P5 are k-Turán-good for k≥4.

Let G=(V,E) be a graph and c:(V∪E)→C be a proper total colouring of G, where C is a set of colours. We call c inclusion-free if for each vertex, the set of colours appearing on the vertex and the incident edges is not a subset of the respective sets of its neighbours. With a probabilistic argument we show that the minimum number of colours for inclusion-free total colouring, denoted by χ⊂″(G), is bounded

In this paper, we study oriented bipartite graphs. In particular, we introduce “bitransitive” graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic bitournaments. As applications, we characterize acyclic bitournaments with Hamiltonian paths, determine the number of non-isomorphic acyclic bitournaments of a given

A set D of vertices in a graph G is called dominating if every vertex of G is either in D or adjacent to a vertex of D. The domination number γ(G) is the minimum size of a dominating set in G, the paired domination number γpr(G) is the minimum size of a dominating set whose induced subgraph admits a perfect matching, and the upper domination number Γ(G) is the maximum size of a minimal dominating set

We study graphs coming from quadratic spaces over finite fields which generalize a recent result given by Bishnoi, Ihringer, and Pepe (2020). More precisely, we study the graph Γ□(n,k,q) defined as follows: the vertex set is the set of k-dimensional quadratic subspaces of a fixed quadratic space (Fqn,x12+⋯+xn−12+λxn2) isometrically isomorphic to x12+⋯+xk2. Here λ is a non-square in Fq, and two vertices

A sequence of vertices in a graph G without isolated vertices is called a total dominating sequence if every vertex v in the sequence has a neighbor which is adjacent to no vertex preceding v in the sequence, and at the end every vertex of G has at least one neighbor in the sequence. Minimum and maximum lengths of a total dominating sequence is the total domination number of G (denoted by γt(G)) and

Let G be a connected undirected graph on n vertices with no loops but possibly multiedges. Given an arithmetical structure (r,d) on G, we describe a construction which associates to it a graph G′ on n−1 vertices and an arithmetical structure (r′,d′) on G′. By iterating this construction, we derive an upper bound for the number of arithmetical structures on G depending only on the number of vertices

A graph drawn on the plane is k-immersed (k≥1) in the plane if each edge is crossed by at most k other edges. By a proper k-immersion of a graph we mean a k-immersion of the graph in the plane such that there is at least one crossing point. Planar graphs having no proper 1-immersions were constructed before, and it was shown that every connected planar graph has a proper 3-immersion (with the exception

Alon and Füredi (1993) showed that the number of hyperplanes required to cover {0,1}n∖{0} without covering 0 is n. We initiate the study of such exact hyperplane covers of the hypercube for other subsets of the hypercube. In particular, we provide exact solutions for covering {0,1}n while missing up to four points and give asymptotic bounds in the general case. Several interesting questions are left

In a recent paper, Müllner and Ryzhikov posed a question about palindromic subwords of finite binary words which can be rephrased as follows: Given four equally long binary words w1,w2,w3,w4 of total length n, what is the size of a longest palindrome p=qqR such that (i) q is a subword of w1w2 and also of (w3w4)R or (ii) q is a subword of w2w3 and also of (w4w1)R? Müllner and Ryzhikov conjectured that

Motivated by an application in condensed matter physics and quantum information theory, we prove that every non-null even-hole-free claw-free graph has a simplicial clique, that is, a clique K such that for every vertex v∈K, the set of neighbours of v outside of K is a clique. In fact, we prove the existence of a simplicial clique in a more general class of graphs defined by forbidden induced subgraphs

A graph G is k-degenerate if each subgraph of G has a vertex of degree at most k. It is known that every simple planar graph with girth 6, or equivalently without 3-, 4-, and 5-cycles, is 2-degenerate. In this work, we investigate for which k every planar graph without 4-, 6-, … , and 2k-cycles is 2-degenerate. We determine that k is 5 and the result is tight since the truncated dodecahedral graph

A graph drawn on the plane is 2-immersed in the plane if each edge is crossed by at most two other edges. By a proper 2-immersion of a graph we mean a 2-immersion of the graph in the plane such that there is at least one crossing point. We consider the class T of all finite graphs triangulating the plane such that the graphs have no loops and multiple edges, the vertices have degree 5 and 6 only, and

Every planar simple graph with n vertices has at least 2n/9 Z5-colorings.

The toughness t(G) of a graph G is a measure of its connectivity that is closely related to Hamiltonicity. Xiaofeng Gu, confirming a longstanding conjecture of Brouwer, recently proved the lower bound t(G)≥ℓ/λ−1 on the toughness of any connected ℓ-regular graph, where λ is the largest nontrivial absolute eigenvalue of the adjacency matrix. Brouwer had also observed that many families of graphs (in

The conjecture, still widely open, posed by Marco Buratti, Peter Horak and Alex Rosa states that a list L of v−1 positive integers not exceeding ⌊v2⌋ is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set {0,1,…,v−1} if and only if, for every divisor d of v, the number of multiples of d appearing in L is at most v−d. In this paper we present new methods that

For any graph G, a set of vertices V is said to be dominating if every vertex of G contains at least one node of G and separating if each vertex v contains a unique neighbour uv∈V that is adjacent to no other vertex of G. If V is both dominating and separating, then V is defined to be an identification code. In this paper, we study strong identification codes with an index r, by imposing the constraint

A bipartite graph with bipartition A and B is said to be Hamilton-laceable if for any u∈A and v∈B there is a Hamilton path joining u and v. It is known that the double generalized Petersen graph DP(n,k) is Hamiltonian and is bipartite if and only if n is even. In this paper we show that the bipartite double generalized Petersen graph DP(n,k) is Hamilton-laceable for n≥4.

In this paper we give two new refinements to the known necessary conditions for the existence of a 4-GDD and list all the feasible group types for 4-GDDs where the number of points v satisfies 31≤v≤50. For v in this range, we obtain solutions for all feasible group types when v≡0 (mod3), and solutions for all feasible types of the form 4s1tw1 except 4811101. We also obtain various other 4-GDDs. These

We show that there exist convex n-gons P and Q such that the largest convex polygon in the Minkowski sum P+Q has size Θ(nlog⁡n). This matches an upper bound of Tiwary.

The weight of an edge e is the degree-sum of its end-vertices. An edge e=uv is an (i,j)-edge if deg(u)≤i and deg(v)≤j. In 1955, Kotzig proved that every 3-connected planar graph contains an edge of weight at most 13. Later, Borodin extended this result to the class of simple planar graphs with minimum degree at least 3. If we consider the class of plane graphs with minimum degree two, the existence