Let p be a prime number, in this paper, we investigate the structures of all constacyclic codes of length ps over the ring Ru2,v2,pm=Fpm[u,v]∕〈u2,v2,uv−vu〉. The units of the ring Ru2,v2,pm can be divided into following five forms: α, λ1=α+δ1uv, λ2=α+γv+δuv, λ3=α+βu+δuv, λ4=α+βu+γv+δuv, where α,β,γ,δ1∈Fpm∗ and δ∈Fpm. We obtain the algebraic structures of all constacyclic codes of length ps over Ru2

An odd [1,b]-factor of a graph G is a spanning subgraph H such that for each vertex v∈V(G), dH(v) is odd and 1≤dH(v)≤b. Let λ3(G) be the third largest eigenvalue of the adjacency matrix of G. For positive integers r≥3 and even n, Lu et al. (2010) proved a lower bound for λ3(G) in an n-vertex r-regular graph G to guarantee the existence of an odd [1,b]-factor in G. In this paper, we improve the bound;

In this paper we are concerned with the classification of the finite groups admitting a bipartite DRR and a bipartite GRR. First, we find a natural obstruction that prevents a finite group from admitting a bipartite GRR. Then we give a complete classification of the finite groups satisfying this natural obstruction and hence not admitting a bipartite GRR. Based on these results and on some extensive

We show there is a bijection between the binary necklaces with n black beads and k white beads and certain (n,k)-codes when n is prime. The main idea is to come up with a new map on necklaces called slime migration.

For given graphs H1,…,Ht, we say that G is Ramsey for H1,…,Ht and we write G⟶(H1,…,Ht), if no matter how one colors the edges of G with t colors, say 1,…,t, there exists a monochromatic copy of Hi in the ith color for some 1≤i≤t. The multicolor Ramsey number r(H1,…,Ht) is the smallest integer n such that the complete graph Kn is Ramsey for (H1,…,Ht). The multicolor size Ramsey number rˆ(H1,…,Ht) is

Gyárfás and Lehel and independently Faudree and Schelp proved that in any 2-coloring of the edges of Kn,n there exists a monochromatic path on at least 2⌈n∕2⌉ vertices, and this is tight. We prove a stability version of this result which holds even if the host graph is not complete; that is, for every γ≫η>0 and n≥n0(γ), if G is a balanced bipartite graph on 2n vertices with minimum degree at least

An orientation of a graph is semi-transitive if it is acyclic, and for any directed path v0→v1→⋯→vk either there is no edge between v0 and vk, or vi→vj is an edge for all 0≤i

We give a short and elementary proof of the fact that for a linear complementary pair (C,D), where C and D are 2-sided ideals in a group algebra, D is uniquely determined by C and the dual code D⊥ is permutation equivalent to C. This includes earlier results of Carlet (2018) and Güneri (2018) on nD cyclic codes which have been proved by subtle and lengthy calculations in the space of polynomials.

For a graph G, let odd(G) and ω(G) denote the number of odd components and the number of components of G, respectively. Then it is well-known that G has a 1-factor if and only if odd(G−S)≤|S| for all S⊂V(G). Also it is clear that odd(G−S)≤ω(G−S). In this paper we characterize a 1-tough graph G, which satisfies ω(G−S)≤|S| for all 0̸≠S⊂V(G), using an H-factor of a set-valued function H:V(G)→{{1},{0,2}}

Denote by λKv the complete graph of order v with multiplicity λ. Let λKv−λKw−λKu be the graph obtained from λKv by the removal of the edges of two vertex disjoint complete multi-subgraphs with multiplicity λ of orders w and u, respectively. When λ is odd, it is shown that there exists a triangle decomposition of λKv−λKw−λKu if and only if v≥w+u+max{u,w}, λv2−u2−w2≡0(mod3) and λ(v−w)≡λ(v−u)≡λ(v−1)≡0(mod2)

An even cycle decomposition of a graph is a partition of its edges into cycles of even length. In 2012, Markström conjectured that the line graph of every 2-connected cubic graph has an even cycle decomposition and proved this conjecture for cubic graphs with oddness at most 2. However, for 2-connected cubic graphs with oddness 2, Markström only considered these graphs with a chordless 2-factor. (A

For any positive integer k and any n-vertex graph G, let ι(G,k) denote the size of a smallest set D of vertices of G such that the graph obtained from G by deleting the closed neighbourhood of D contains no k-clique. Thus, ι(G,1) is the domination number of G. We prove that if G is connected, then ι(G,k)≤nk+1 unless G is a k-clique, or k=2 and G is a 5-cycle. The bound is sharp. The case k=1 is a classical

A spanning circuit in a graph is defined as a closed trail visiting each vertex of the graph. A compatible spanning circuit in an edge-colored graph refers to a spanning circuit in which each pair of edges traversed consecutively along the spanning circuit has distinct colors. As two extreme cases, sufficient conditions for the existence of compatible Hamilton cycles and compatible Euler tours have

The minimum leaf number of a connected non-hamiltonian graph G is the number of leaves of a spanning tree of G with the fewest leaves among all spanning trees of G. Based on this quantity, Wiener introduced leaf-stable and leaf-critical graphs, concepts which generalise hypotraceability and hypohamiltonicity. In this article, we present new methods to construct leaf-stable and leaf-critical graphs

A finite group is said to be metacyclic if it has a cyclic normal subgroup N such that G∕N is cyclic. A code over a finite field F is called metacyclic if it is a left ideal in the group algebra FG, with G a metacyclic group. Metacyclic codes are generalizations of dihedral codes, and can be constructed as quasi-cyclic codes with an extra automorphism. In this paper, we prove that metacyclic codes

Consider a sum Sn=viε1+⋯+vnεn, where (vi)i=1n are non-zero vectors in Rd and (εi)i=1n are independent Rademacher random variables (i.e., P(εi=±1)=12). The classical Littlewood–Offord problem asks for the best possible upper bound for supxP(Sn=x). In this paper we consider a non-uniform version of this problem. Namely, we obtain the optimal bound for P(Sn=x) in terms of the length of the vector x∈Rd

For a directed graph G, a vertex x is a king if every other vertex can be reached from x by a directed path of length at most 2 and is a serf if x can be reached from every other vertex by a directed path of length at most 2. A tournament T with vertex set V is called X-hamiltonian crossing if every hamiltonian path must have one end in X and the other in X̄=V−X. Our main results are the following:

For any sequence u, the extremal function Ex(u,j,n) is the maximum possible length of a j-sparse sequence with n distinct letters that avoids u. We prove that if u is an alternating sequence abab… of length s, then Ex(u,j,n)=Θ(sn2) for all j≥2 and s≥n, answering a question of Wellman and Pettie (2018) and extending the result of Roselle and Stanton that Ex(u,2,n)=Θ(sn2) for any alternation u of length

We are given a set A of buyers, a set B of houses, and for each buyer a preference list, i.e., an ordering of the houses. A house allocation is an injective mapping τ from A to B, and τ is strictly better than another house allocation τ′≠τ if for every buyer i, τ′(i) does not come before τ(i) in the preference list of i. A house allocation is Pareto optimal if there is no strictly better house allocation

A complex square matrix is called a ray nonsingular matrix (RNS matrix) if its ray pattern implies that it is nonsingular. A matrix M=I−A(W) is called a cycle tree matrix if the adjacency structure of the cycles in the arc-weighted digraph W (with no multi-arcs or loops), which is described by the cycle graph of W, is a tree. In this paper, it is shown that if there is no positive cycle in W, then

Let d,n∈Z+ such that 1≤d≤n. A d-code C⊂Fqn×n is a subset of order n square matrices with the property that for all pairs of distinct elements in C, the rank of their difference is greater than or equal to d. A d-code with as many as possible elements is called a maximum d-code. The integer d is also called the minimum distance of the code. When d

We present new results on the traceability of claw-free graphs. In particular, we consider sufficient minimum degree and degree sum conditions that imply that these graphs admit a Hamilton path, unless they have a small order or they belong to well-defined classes of exceptional graphs. Our main result implies that a 2-connected claw-free graph G of sufficiently large order n with δ(G)≥3 is traceable

In 1956, Alder conjectured that qd(n)−Qd(n)≧0, where qd(n) and Qd(n) are the number of partitions of n into parts differing by at least d and the number of partitions of n into parts which are congruent to ±1 (mod d+3), respectively. It took more than 50 years to complete the proof and the first breakthrough was made by Andrews in 1971, who proved that the conjecture holds for d=2r−1 (r≧4). In this

An arithmetical structure on a finite, connected graph G is a pair of vectors (d,r) with positive integer entries for which (diag(d)−A)r=0, where A is the adjacency matrix of G and where the entries of r have no common factor. The critical group of an arithmetical structure is the torsion part of the cokernel of (diag(d)−A). In this paper, we study arithmetical structures and their critical groups

A frieze is an array of numbers obeying the unimodular rule. Coxeter showed that a frieze with integer entries corresponds to a triangulation. Recently, Holm and Jørgenson introduced friezes of type Λp which correspond to p-angulations of a polygon. In this paper we explore the connection between these two types of friezes; in particular we show that the friezes of type Λp for p=4 and p=6 contain integral

Buch and Rimányi proved a formula for a specialization of double Grothendieck polynomials based on the Yang–Baxter equation related to the degenerate Hecke algebra. A geometric proof was found by Yong and Woo by constructing a Gröbner basis for the Kazhdan–Lusztig ideals. In this note, we give an elementary proof for this formula by using only divided difference operators.

In this paper, we study the relationship between t-core Young diagrams and t-core shifted Young diagrams. In particular, we obtain a characterization of t-core shifted Young diagram. Furthermore, we apply the characterization to deduce the number and the largest size of (t,t+1)-core shifted Young diagrams.

Given a graph G, denote by Δ(G) and χ′(G) the maximum degree and the chromatic index of G, respectively. A simple graph G is called edge-Δ-critical if Δ(G)=Δ, χ′(G)=Δ+1 and χ′(H)≤Δ for every proper subgraph H of G. We prove that every edge-Δ-critical graph of order n with maximum degree at least 2n3+12 is Hamiltonian.

This paper will primarily present a method of proving generating function identities for partitions from linked partition ideals. The method we introduce is built on a conjecture by George Andrews and that those generating functions satisfy some q-difference equations. We will come up with the generating functions of partitions in the Kanade–Russell conjectures to illustrate the effectiveness of this

In 2012, Momihara et al. (2012) showed that the 2-design formed by the planes (2-flats) in AG(2n,3) can be decomposed into more subdesigns than a previously known decomposition. They restricted the group stabilizing the resulting subdesigns to the affine general linear group AGL(1,32n) and its subgroups, and then gave the best decomposition in the sense that the total number of subdesigns is maximum

Euler showed that the number of partitions of n into distinct parts equals the number of partitions of n into odd parts. This theorem was generalized by Glaisher and further by Franklin. Recently, Beck made three conjectures on partitions with restricted parts, which were confirmed analytically by Andrews and Chern and combinatorially by Yang. Analogous to Euler’s partition theorem, it is known that

Let G=(V,E) be a simple graph and ϕ:E(G)→{1,2,…,k} be a proper k-edge coloring of G. We say that ϕ is neighbor sum (set) distinguishing if for each edge uv∈E(G), the sum (set) of colors taken on the edges incident with u is different from the sum (set) of colors taken on the edges incident with v. The smallest k such that G has a neighbor sum (set) distinguishing k-edge coloring is called the neighbor

Two decades ago, Chauve, Dulucq and Guibert showed that the number of rooted trees on the vertex set [n+1] in which exactly k children of the root are lower-numbered than the root is nknn−k. Here I give a simpler proof of this result.

We provide a probabilistic approach using renewal theory to derive some novel identities involving Eulerian numbers and uniform B-splines. The renewal perspective leads to a unified treatment for the normalized binomial coefficients and the normalized Eulerian numbers when studying their limits of sums, as well as their associated distributions — the binomial distributions (Bernstein polynomials) and

In this paper we give a simpler proof of a deep theorem proved by Pellikan, Shen and van Wee that all linear codes are weakly algebraic-geometric using a theorem of B. Poonen.

Constructing minimal linear codes is an interesting research topic due to their applications in coding theory and cryptography. Ashikhmin and Barg pointed out that wmin∕wmax>(q−1)∕q is a sufficient condition for a linear code over the finite field Fq to be minimal, where wmin and wmax respectively denote the minimum and maximum nonzero weights in a code. However, only a few families of minimal linear

The theory of matroids has been generalized to oriented matroids and, recently, to arithmetic matroids. We want to give a definition of “oriented arithmetic matroid” and prove some properties like the “uniqueness of orientation”.

The goal of this paper is to advertise the method of gauge transformations (aka holographic reduction, reparametrization) that is well-known in statistical physics and computer science, but less known in combinatorics. As an application of it we give a new proof of a theorem of A. Schrijver asserting that the number of Eulerian orientations of a d–regular graph on n vertices with even d is at least

We give an example of a graphon such that there is no equivalent graphon with a degree function that is (weakly) increasing.

In this article, we state and prove a general criterion which prevents some groups from acting properly on finite-dimensional CAT(0) cube complexes. As an application, we show that, for every non-trivial finite group F, the lamplighter group F≀F2 over a free group does not act properly on a finite-dimensional CAT(0) cube complex (although it acts properly on an infinite-dimensional CAT(0) cube complex)

In 1963, Corrádi and Hajnal settled a conjecture of Erdős by showing that every graph on at least 3r vertices with minimum degree at least 2r contains a collection of r disjoint cycles, and in 2008, Finkel proved that every graph with at least 4s vertices and minimum degree at least 3s contains a collection of s disjoint chorded cycles. The same year, a generalization of this theorem was conjectured

Let G be a graph other than a path. The m-iterated line graph of a graph G is Lm(G)=L(Lm−1(G)). where L1(G) denotes the line graph L(G) of G. Define the Hamiltonian index h(G) of G to be the smallest integer m such that Lm(G) contains a Hamiltonian cycle. For a connected graph set H, G is said to be H-free if G does not contain H as an induced subgraph for all H∈H. In this paper, we characterize all

The chromatic- edge-stability number esχ(G) of a graph G is the minimum number of edges whose removal results in a spanning subgraph G′ with χ(G′)=χ(G)−1. Edge-stability critical graphs are introduced as the graphs G with the property that esχ(G−e)

In this paper it is shown that any partial Latin square of order n can be embedded in a Latin square of order at most 16n2 which has at least 2n mutually orthogonal mates. Further, for any t⩾2, it is shown that a pair of orthogonal partial Latin squares of order n can be embedded in a set of t mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to n. A consequence of the constructions

Let c1,c2,…,ck be non-negative integers. A graph G is (c1,c2,…,ck)-colorable if the vertex set of G can be partitioned into k sets V1,V2,…,Vk, such that G[Vi], the subgraph induced by Vi, has maximum degree at most ci for i∈[k]. Wang and Xu (2015) posed some open problems, one of which is that every planar graph without adjacent triangles is (1,1,1)-colorable. In this paper, we prove that every planar

In 1986, Thomassen conjectured that every 4-connected line graph is hamiltonian. The conjecture is still wide open, and, as a possible approach to it, many statements that are equivalent or related to it have been studied. In this paper, we extend the statement to the class of line graphs of 3-hypergraphs, and generalize it to Tutte cycles and paths (note that a line graph of a 3-hypergraph is K1,4-free

Cyclic sieving is a well-known phenomenon where certain interesting polynomials, especially q-analogues, have useful interpretations related to actions and representations of the cyclic group. We propose a definition of sieving for an arbitrary group G and study it for the dihedral group I2(n) of order 2n. This requires understanding the generators of the representation ring of the dihedral group.

Zykov showed in 1949 that among graphs on n vertices with clique number ω(G)≤ω, the Turán graph Tω(n) maximizes not only the number of edges but also the number of copies of Kt for each size t. The problem of maximizing the number of copies of Kt has also been studied within other classes of graphs, such as those on n vertices with maximum degree Δ(G)≤Δ. We combine these restrictions and investigate

We introduce the poset of mesh patterns, which generalizes the permutation pattern poset. We fully classify the mesh patterns for which the interval [10̸,m] is non-pure, where 10̸ is the unshaded singleton mesh pattern. We present some results on the Möbius function of the poset, and show that μ(10̸,m) is almost always zero. Finally, we introduce a class of disconnected and non-shellable intervals

The p-rank of a Steiner quadruple system B is the dimension of the linear span of the set of characteristic vectors of blocks of B over GF(p). We derive a formula for the number of different Steiner quadruple systems of order v and given 2-rank r, r

Let D be a symmetric 2-(v,k,λ) design with λ prime and G be a flag-transitive and point-primitive automorphism group of D. Then G must be of affine or almost simple type.

The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph and a matching. We prove that this conjecture is true for connected cubic graphs with a 2-factor consisting of three cycles.

A strong edge-coloring of a graph G is an edge-coloring such that any two edges on a path of length three receive distinct colors. We denote the strong chromatic index by χs′(G) which is the minimum number of colors that allow a strong edge-coloring of G. Erdős and Nešetřil conjectured in 1985 that the upper bound of χs′(G) is 54Δ2 when Δ is even and 14(5Δ2−2Δ+1) when Δ is odd, where Δ is the maximum

Let G be a plane graph. Two edges are said to be facially adjacent if they are consecutive on the boundary walk of a face of G. We call G facially edge–face k-colorable if there is a mapping from E(G)∪F(G) to a k color set so that any two facially adjacent edges, adjacent faces, and incident edge and face receive distinct colors. The facial edge–face chromatic number of G, denoted by χ̄ef(G), is defined

Let E be a set of solids (hyperplanes) in PG(4,q), q even, q>2, such that every point of PG(4,q) lies in either 0, 12(q3−q2) or 12q3 solids of E, and every plane of PG(4,q) lies in either 0, 12q or q solids of E. This article shows that E is either the set of solids that are disjoint from a hyperoval, or the set of solids that meet a non-singular quadric Q(4,q) in an elliptic quadric.

This paper presents a formula for the cardinality of a class of non-linear error correcting codes for Balanced Adjacent Deletions that are provided as an extension of standard deletion from the point of the view of Weyl groups. Furthermore, we show that the cardinality is approximately optimal over any single BAD correcting codes. In other words, the ratio of the cardinality of the code and that of

A planar support for a hypergraph is a planar graph that captures the structural properties of the hypergraph. In this paper, we show, using elementary techniques, that a wide class of geometric hypergraphs of planar non-piercing regions admit a planar support. We also discuss some direct consequences of our results.

The intersection enumerator and the Jacobi polynomial in an arbitrary genus for a binary code are introduced. Adding the weight enumerator into our discussion, we give the explicit relations among them and give some of their basic properties.

Using an algebraic characterization of circle graphs, Bouchet proved in 1999 that if a bipartite graph G is the complement of a circle graph, then G is a circle graph. We give an elementary proof of this result.

Let Σ be a signed graph where two edges joining the same pair of vertices with opposite signs are allowed. The zero-free chromatic number χ∗(Σ) of Σ is the minimum even integer 2k such that G admits a proper coloring f:V(Σ)↦{±1,±2,…,±k}. The zero-free list chromatic number χl∗(Σ) is the list version of zero-free chromatic number. Σ is called zero-free chromatic-choosable if χl∗(Σ)=χ∗(Σ). We show that