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In this paper, we study some topics concerning the additive decompositions of the set D k of all k th power residues modulo a prime p. For example, given a positive integer k ≥ 2 , we prove that lim x → + ∞ ⁡ B (x) π (x) = 0 , where π (x) is the number of primes p ≤ x and B (x) denotes the cardinality of the set { p ≤ x : p ≡ 1 (mod k) ; D k has a non-trivial 2-additive decomposition }. [ABSTRACT FROM AUTHOR]

In this paper, we consider 2 k -cycle decomposition of K m × K n and directed 2 k -cycle decompositions of ( K m ∘ K ¯ n ) ∗ and ( K m × K n ) ∗ , where ∘ and × denote the wreath product and tensor product of graphs, respectively. Using the results obtained here, we prove that for m , n ≥ 3 , the obvious necessary conditions for the existence of a C 2 k -decomposition of K m × K n are sufficient whenever k ∈ { p , 2 ℓ } , where p is a prime and ℓ ≥ 2 . Also, we show that the necessary conditions for the existence of C → 2 p -decompositions of ( K m ∘ K ¯ n ) ∗ and ( K m × K n ) ∗ are sufficient whenever p is a prime, where C → 2 p denotes the directed cycle of length 2 p . [ABSTRACT FROM AUTHOR]

We show that for every prime number p , a 3-connected non-uniform G F ( p ) -representable matroid can have a clone set of size at most p − 2 . [ABSTRACT FROM AUTHOR]

Fix an odd prime ℓ and let G be the poset of isomorphism classes of finite abelian ℓ -groups, ordered by inclusion. If ξ : G → R ≥ 0 is a discrete probability distribution on G and A ∈ G , define the A th moment of ξ to be ∑ B ∈ G | Surj ( B , A ) | ξ ( B ) . The question of determining conditions that ensure ξ is completely determined by its moments has been of recent interest in many problems of Cohen–Lenstra type. Furthermore, recovering ξ from its moments requires a new Möbius-type inversion formula on G . In this paper, we define this function, relate it to the classical Möbius function on the poset of subgroups of a fixed group, and prove two theorems about when it vanishes. As one corollary of these theorems, we obtain an analog of Hall’s theorem on the vanishing of the classical Möbius function. As another, we obtain an infinite family of pairs of groups on which the classical Möbius function vanishes; obtaining such pairs is a group-theoretic topic of recent interest. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for any 1 / 2 < t < 1 , there exists a positive integer N 0 depending on t such that for any n 0 ≥ N 0 , squares of sidelength f (n) − t for n ≥ n 0 can be packed with disjoint interiors into a square of area ∑ n = n 0 ∞ f (n) − 2 t , if the function f satisfies some suitable conditions. The main theorem (Theorem 1.1) is a generalization of Tao's theorem in [15] , which argued the case f (n) = n. As corollaries, we prove that there are such packings of squares when f (n) represents the n th element of either an arithmetic progression or the set of prime numbers. In these cases, we give effective lower bounds for N 0 with respect to t. Furthermore, we consider the case that f (n) represents the n th element of the set of twin primes and prove that squares of sidelength f (n) − t for n ≥ n 0 can be packed with disjoint interiors into a slightly larger square than theoretically expected. [ABSTRACT FROM AUTHOR]

Bierbrauer (2012) developed the theory of q -linear cyclic codes over ( F q ) m and he obtained a parametric description of such codes by cyclotomic cosets. Recently, Cao et al. (2015) obtained the structure of cyclic additive codes over the Galois ring GR ( p ℓ , m ) , where m is a prime integer. Let R be a finite commutative ring and R n = R [ x ] ∕ 〈 x n − 1 〉 . In this paper, we generalize the theory of F q -linear codes over vector spaces to R -linear codes over free R -algebras (free as R -module). We call these codes, R -additive codes. We introduce a one-to-one correspondence between the classes of cyclic R -additive code and the classes of R n -linear code. Using the structure of R n -linear codes, we present the structure of cyclic R -additive codes, where R is a chain ring. Among other results, q -linear cyclic codes over ( F q ) m are described by ring-theoretic facts, and the structure of cyclic additive codes over the Galois ring GR ( p ℓ , m ) is given for an arbitrary integer m , not necessarily a prime number. [ABSTRACT FROM AUTHOR]

Let p be a prime number and K be the finite field of p elements, i.e. K = G F ( p ) . Further let G be an elementary abelian p -group of order p m . Then the group algebra K [ G ] is modular. We consider K [ G ] as an ambient space and the ideals of K [ G ] as linear codes. A basis of a linear space is called visible, if there exists a member of the basis with the minimum (Hamming) weight of the space. The group algebra approach enables us to find some linear codes with a visible basis in the Jacobson radical of K [ G ] . These codes can be generated by “monomials” (Drensky & Lakatos, 1989). For p > 2 , some of our monomial codes have better parameters than the Generalized Reed–Muller codes. In the last part of the paper we determine the automorphism groups of some of the introduced codes. [ABSTRACT FROM AUTHOR]

The adjacency spectrum of a graph Γ , which is denoted by S p e c ( Γ ) , is the multiset of eigenvalues of its adjacency matrix. We say that two graphs Γ and Γ ′ are cospectral if S p e c ( Γ ) = S p e c ( Γ ′ ) . In this paper for each prime number p , p ≥ 23 , we construct a large family of cospectral non-isomorphic Cayley graphs over the dihedral group of order 2 p . [ABSTRACT FROM AUTHOR]

Let F q be a finite field of order q = p e , where p is a positive prime. For m ≥ 1 , let P and L be two copies of F q m + 1 . To each m -tuple g = ( g 2 , … , g m + 1 ) of polynomials in F q [ x , y ] , we consider the bipartite graph W q ( g ) . The vertex set V of W q ( g ) is P ∪ L . The edge set E of W q ( g ) consists of ( p , l ) ∈ P × L satisfying p 2 + l 2 = g 2 ( p 1 , l 1 ) , p 3 + l 3 = g 3 ( p 1 , l 1 ) , … , p m + 1 + l m + 1 = g m + 1 ( p 1 , l 1 ) , where p = ( p 1 , p 2 , … , p m + 1 ) ∈ P and l = ( l 1 , l 2 , … , l m + 1 ) ∈ L . W q ( g ) is called linearized Wenger graph when g = ( x y , x p y , … , x p m − 1 y ) . In this paper, we determine the eigenvalues of linearized Wenger graph and their multiplicities in the case of m < e , which is an open problem put forward by Cao et al. (2015). [ABSTRACT FROM AUTHOR]

A dihedral SQS is an SQS admitting a point-regular dihedral automorphism group. Some constructions of dihedral SQSs are established in this paper. We also prove that for v ≡ 2 , 4 ( m o d 6 ) , there is a dihedral SQS ( v ) if there is a dihedral H ( p , 2 , 4 , 3 ) design or an S -cyclic SQS ( 2 p ) for each odd prime divisor p of v . As a corollary, we enlarge the set of known values of v for which there exists a dihedral SQS ( v ) . [ABSTRACT FROM AUTHOR]

An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ( 2 -colored) cycles. The acyclic chromatic index of a graph G , denoted by a ′ ( G ) , is the least integer k such that G admits an acyclic edge-coloring using k colors. Let Δ = Δ ( G ) denote the maximum degree of a vertex in a graph G . A complete bipartite graph with n vertices on each side is denoted by K n , n . Basavaraju, Chandran and Kummini proved that a ′ ( K n , n ) ≥ n + 2 = Δ + 2 when n is odd. Basavaraju and Chandran provided an acyclic edge-coloring of K p , p using p + 2 colors and thus establishing a ′ ( K p , p ) = p + 2 = Δ + 2 when p is an odd prime. The main tool in their approach is perfect 1 -factorization of K p , p . Recently, following their approach, Venkateswarlu and Sarkar have shown that K 2 p − 1 , 2 p − 1 admits an acyclic edge-coloring using 2 p + 1 colors which implies that a ′ ( K 2 p − 1 , 2 p − 1 ) = 2 p + 1 = Δ + 2 , where p is an odd prime. In this paper, we generalize this approach and present a general framework to possibly get an acyclic edge-coloring of K n , n which possesses a perfect 1 -factorization using n + 2 = Δ + 2 colors. In this general framework, using number theoretic techniques, we show that K p 2 , p 2 admits an acyclic edge-coloring with p 2 + 2 colors and thus establishing a ′ ( K p 2 , p 2 ) = p 2 + 2 = Δ + 2 when p is an odd prime. [ABSTRACT FROM AUTHOR]

Recently, linear codes with a few weights have been constructed and extensively studied. In this paper, for an odd prime p , we determine the complete weight enumerator of two classes of p -ary linear codes constructed from defining set. Our results show that the codes have at most seven weights and may have applications in secret sharing schemes. [ABSTRACT FROM AUTHOR]

Let F q 2 be the finite field of q 2 elements, where q is a power of a prime number, and let D 2 n = 〈 x , y | x n = 1 , y 2 = 1 , y x y = x n − 1 〉 be the dihedral group of 2 n elements. Left ideals of the group algebra F q 2 [ D 2 n ] are known as left dihedral codes over F q 2 of length 2 n , and abbreviated as left D 2 n -codes. Let gcd (n , q) = 1. In this paper, we give an explicit representation for the Hermitian dual code and the Hermitian hull of every left D 2 n -code over F q 2 . On this basis, we determine all distinct Hermitian self-dual left D 2 n -codes, Hermitian linear complementary dual (LCD) left D 2 n -codes, and Hermitian self-orthogonal left D 2 n -codes over F q 2 , respectively. Then we provide an explicit representation and a precise enumeration for these three subclasses of left D 2 n -codes. As an application, we provide several illustrative examples for obtaining Hermitian self-dual and Hermitian LCD left D 2 n -codes respectively. [ABSTRACT FROM AUTHOR]

A regular balanced Cayley map (RBCM for short) on a finite group Γ is an embedding of a Cayley graph on Γ into a surface, with some special symmetric property. People have classified RBCM’s for cyclic, dihedral, generalized quaternion, dicyclic, and semi-dihedral groups. In this paper we classify RBCM’s on the group PSL ( 2 , p ) for each prime number p > 3 . [ABSTRACT FROM AUTHOR]

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Let n be a product of three primes. The problem on the classification of the tetravalent half-arc-transitive graphs of order n has been considered by Xu (1992), Feng et al. (2007) and Wang and Feng (2010), and it was solved for the cases where n is a prime cube or twice a product of two primes. In this paper, we solve this problem for the remaining cases. In particular, there exist some families of these graphs which have a solvable automorphism group but are not metacirculants. [ABSTRACT FROM AUTHOR]

This paper characterizes the automorphism group G of the orientably-regular embeddings of simple graphs of order prime-cube p 3 . Our main result will be a starting point for classifying all such embeddings. Moreover, by using some known results, a partial classification is given, when G contains a Sylow p -subgroup of order p 5 . [ABSTRACT FROM AUTHOR]

In this paper we classify the reflexible and chiral regular oriented maps with p faces of valency n , and then we compute the asymptotic behaviour of the reflexible to chiral ratio of the regular oriented maps with p faces. The limit depends on p and for certain primes p we show that the limit can be 1, greater than 1 and less than 1. In contrast, the reflexible to chiral ratio of regular polyhedra (which are regular maps) with Suzuki automorphism groups, computed by Hubard and Leemans (2014), has produced a nill asymptotic ratio. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements, l an odd prime, and t , v positive integers such that l v | ( q − 1 ) and gcd ( t , l ) = 1 . In this paper, we give a combinatorial result and use it to determine the weight distribution of a cyclic code of length t l m with t | ( q − 1 ) , which is an open question in Yang et al. (2013). Moreover we compute the weight distributions of cyclic codes of length t l m , where q ≡ 3 ( mod 4 ) and t = 4 or 8 . [ABSTRACT FROM AUTHOR]

Let Γ be an abelian group and g ≥ h ≥ 2 be integers. A set A ⊂ Γ is a C h [ g ] -set if given any set X ⊂ Γ with | X | = h , and any set { k 1 , … , k g } ⊂ Γ , at least one of the translates X + k i is not contained in A . For any g ≥ h ≥ 2 , we prove that if A ⊂ { 1 , 2 , … , n } is a C h [ g ] -set in Z , then | A | ≤ ( g − 1 ) 1 / h n 1 − 1 / h + O ( n 1 / 2 − 1 / 2 h ) . We show that for any integer n ≥ 1 , there is a C 3 [ 3 ] -set A ⊂ { 1 , 2 , … , n } with | A | ≥ ( 4 − 2 / 3 + o ( 1 ) ) n 2 / 3 . We also show that for any odd prime p , there is a C 3 [ 3 ] -set A ⊂ F p 3 with | A | ≥ p 2 − p , which is asymptotically best possible. Using the projective norm graphs from extremal graph theory, we show that for each integer h ≥ 3 , there is a C h [ h ! + 1 ] -set A ⊂ { 1 , 2 , … , n } with | A | ≥ ( c h + o ( 1 ) ) n 1 − 1 / h . A set A is a weak C h [ g ] -set if we add the condition that the translates X + k 1 , … , X + k g are all pairwise disjoint. We use the probabilistic method to construct weak C h [ g ] -sets in { 1 , 2 , … , n } for any g ≥ h ≥ 2 . Lastly we obtain upper bounds on infinite C h [ g ] -sequences. We prove that for any infinite C h [ g ] -sequence A ⊂ N , we have A ( n ) = O ( n 1 − 1 / h ( log n ) − 1 / h ) for infinitely many n , where A ( n ) = | A ∩ { 1 , 2 , … , n } | . [ABSTRACT FROM AUTHOR]

We consider a problem due to Mariusz Meszka similar to the well-known conjecture of Marco Buratti. Does there exist a near-1-factor in the complete graph on Z p , p is an odd prime, whose set of edge-lengths equals a given multiset L ? We establish several sufficient conditions for the answer to be yes. [ABSTRACT FROM AUTHOR]

For integer k ≥ 2 and prime power q , an algebraic bipartite graph D ( k , q ) of girth at least k + 4 was introduced by Lazebnik and Ustimenko (1995). Füredi et al. (1995) shown that the girth of D ( k , q ) is equal to k + 5 if k is odd and q is a prime power of form 1 + n ( k + 5 ) / 2 and, conjectured further that D ( k , q ) has girth k + 5 for all odd k and all q ≥ 4 . In this paper, we show that this conjecture is true when ( k + 5 ) / 2 is a power of the characteristic of F q . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the explicit expressions for the generating idempotents, check polynomials and the parameters of all minimal cyclic codes of length over are obtained, where is an odd prime different from the characteristic of , and are positive integers with , and . Our results generalize the main results in Pruthi and Arora (1997) and Arora and Pruthi (1999), which considered the cases and respectively. We propose an approach different from those in Pruthi and Arora (1997) and Arora and Pruthi (1999) to obtain the generating idempotents. [Copyright &y& Elsevier]

Abstract: The well-known middle levels problem is to find a Hamiltonian cycle in the graph induced from the binary Hamming graph by the words of weight or . In this paper we define the -analog of the middle levels problem. Let and let be a power of a prime number. Consider the set of -dimensional subspaces and the set of -dimensional subspaces of . Can these subspaces be ordered in a way that for any two adjacent subspaces and , either or ? A construction method which yields many Hamiltonian cycles for any given and is presented. [Copyright &y& Elsevier]

Abstract: The paper presents classifications of edge-transitive cyclic regular covers of the complete graphs and , and arc-transitive cyclic regular covers of the complete graph . Two new infinite families of transitive graphs of valency and are found. As an application, tetravalent edge-transitive graphs of order with a prime are classified. [Copyright &y& Elsevier]

Abstract: In a graph , a module is a vertex subset such that every vertex outside is adjacent to all or none of . A graph is prime if , the single-vertex sets, and are the only modules in . A prime graph is -minimal if there is some -set of vertices such that no proper induced subgraph of containing is prime. Cournier and Ille in 1998 characterized the -minimal and -minimal graphs. We characterize -minimal triangle-free graphs. As a corollary, we show that there are exactly nonisomorphic -minimal triangle-free -vertex graphs when , where denotes the nearest integer to . [Copyright &y& Elsevier]

Abstract: The finite Figueroa planes are non-Desarguesian projective planes of order for all prime powers , constructed algebraically in 1982 by Figueroa, and Hering and Schaeffer, and synthetically in 1986 by Grundhöfer. All Figueroa planes of finite square order are shown to possess a unitary polarity by de Resmini and Hamilton in 1998, and hence admit unitals. Hui and Wong (2012) have shown that these polar unitals do not satisfy a necessary condition, introduced by Wilbrink in 1983, for a unital to be classical, and hence they are not classical. In this article we introduce and make use of a new alternative synthetic description of the Figueroa plane and unital to demonstrate the existence of O’Nan configurations, thus providing support to Piper’s conjecture (1981). [Copyright &y& Elsevier]

Abstract: For every prime power , where , and a prime dividing , we construct a self-orthogonal code and a self-dual code over the field of order . The construction involves Paley graphs and the constructed and codes admit an automorphism group of the Paley graph of order . If is a prime and , where is a non-negative integer, then the self-dual code is equivalent to a Pless symmetry code. In that sense we can view this class of codes as a generalization of Pless symmetry codes. For and we get a self-dual code whose words of minimum weight form a 3-(20, 8, 28) design. [Copyright &y& Elsevier]

Abstract: Given a graph , a subset of is a module of if for each , is adjacent to all the elements of or to none of them. A graph is prime if and the only modules of are , , and singleton vertex sets. Given a prime induced subgraph , we introduce a digraph that yields a necessary and sufficient condition for to be prime. [Copyright &y& Elsevier]

Abstract: In Matula (1968), D.W. Matula described a bijection between and the set of rooted trees; the number is called the Matula number of the rooted tree. The Gutman–Ivić–Matula (GIM) function computes the number of edges of the tree with Matula number . Since there is a prefix-free code for the set of prime numbers such that the codelength of each prime is , we show how some results about the GIM function can be obtained trivially from coding theorems. [Copyright &y& Elsevier]

Abstract: A graph is called -arc-regular with if acts regularly on its arc set, while is called arc-regular if . J.X. Zhou and Y.Q. Feng [Cubic one-regular graphs of order twice a square-free integer, Sci. China Ser. A 51 (2008) 1093–1100] proved that there is no cubic arc-regular graph of order four times an odd square-free integer. In this paper, we shall generalize this result by showing that there is no arc-regular -valent graph of order four times an odd square-free integer for each odd prime . Moreover, we prove that there are exactly two specific infinite families of -arc-regular graphs with a proper subgroup of . [Copyright &y& Elsevier]

For a finite group G and an inverse closed subset S ⊆ G ∖ { e } , the Cayley graph X (G , S) has vertex set G and two vertices x , y ∈ G are adjacent if and only if x y − 1 ∈ S. Two graphs are called cospectral if their adjacency matrices have the same spectrum. Let p ≥ 3 be a prime number and T 4 p be the dicyclic group of order 4 p. In this paper, with the help of the characters from representation theory, we construct a large family of pairwise non-isomorphic and cospectral Cayley graphs over the dicyclic group T 4 p with p ≥ 23 , and find several pairs of non-isomorphic and cospectral Cayley graphs for 5 ≤ p ≤ 19. [ABSTRACT FROM AUTHOR]

Dinh et al. (2018) [10] obtained all self-dual constacyclic codes of length p s over R 2 = F p m + u F p m , where p is a prime number and u 2 = 0. In this paper, we determine the structure of (Euclidean) dual of some special skew cyclic codes of length p s over R 2 , and establish all of them which are self-dual. As a special case, we conclude all of results appeared in the above paper for cyclic codes. [ABSTRACT FROM AUTHOR]

Abstract: Two sets and of non-negative integers are called additive complements, if their sum contains all sufficiently large integers. Let and be the counting functions of and . Up to now, all researches on additive complements are under the assumption that and are infinite sets. In this paper, we consider the case that is a finite set. This is very different from the infinite case. [Copyright &y& Elsevier]

Abstract: In this article a complete classification of tetravalent arc-transitive graphs of order twice a product of two primes is given. [Copyright &y& Elsevier]

Abstract: Suppose is a finite group, such that , where  is prime. We show that if is any generating set of , then there is a Hamiltonian cycle in the corresponding Cayley graph . [Copyright &y& Elsevier]

Abstract: Let be a prime number and be the cyclic group of order . A 3-coloring of is rainbow-free for some equation if it contains no rainbow solution of the equation. In  Jungić et al. (2003) proved that every 3-coloring of , with the cardinality of the smallest color class greater than four, has a rainbow solution of “almost” all linear equations in three variables in . In this work we handle the “small” cases and give a structural description of rainbow-free colorings for the particular case of , which includes the Schur equation. [Copyright &y& Elsevier]

Abstract: A cellular embedding of a connected graph is said to be Hamiltonian if every face of the embedding is bordered by a Hamiltonian cycle (a cycle containing all the vertices of ) and it is an -gonal embedding if every face of the embedding has the same length . In this paper, we establish a theory of generalized Cayley maps, including a new extension of voltage graph techniques, to show that for each even there exists a Hamiltonian embedding of such that the embedding is a Cayley map and that there is no -gonal Cayley map of if is a prime. In addition, we show that there is no Hamiltonian Cayley map of if , an odd prime and . [Copyright &y& Elsevier]

Abstract: A graph has a representation modulo if there exists an injective map such that vertices and are adjacent if and only if is relatively prime to . The representation number is the smallest such that has a representation modulo . Following earlier work on stars, we study representation numbers of complete bipartite graphs and more generally complete multipartite graphs. [Copyright &y& Elsevier]

Abstract: A graph is said to be -prime if, whenever it is a subgraph of a nontrivial Cartesian product graph, it is a subgraph of one of the factors. A diagonalized Cartesian product is obtained from a Cartesian product graph by connecting two vertices of maximal distance by an additional edge. We show there that a diagonalized product of -prime graphs is again -prime. Klavžar et al. [S. Klavžar, A. Lipovec, M. Petkovšek, On subgraphs of Cartesian product graphs, Discrete Math. 244 (2002) 223–230] proved that a graph is -prime if and only if it admits a nontrivial path--coloring. We derive here a characterization of all path--colorings of Cartesian products of -prime graphs. [Copyright &y& Elsevier]

Abstract: Let and be the largest order of a Cayley graph of a cyclic and an Abelian group, respectively, of diameter 2 and a given degree . There is an obvious upper bound of the form . We prove a number of lower bounds on both quantities for certain infinite sequences of degrees related to primes and prime powers, the best being and . We also offer a result for Cayley graphs of metacyclic groups for general degree and diameter. [Copyright &y& Elsevier]

Abstract: Let be a Hadamard -design. Suppose that the prime divides , but that does not divide . A result of Klemm implies that every residual design of has -rank at least . Also, every derived design of has -rank at least if . We show that when is a skew Hadamard design, the -ranks of the residual and derived designs are at least even if divides or . We construct infinitely many examples where the -rank is exactly . [Copyright &y& Elsevier]

Abstract: Let denote the number of times the prime number appears in the prime factorization of the integer . The following result is proved: If there is a perfect 1-error correcting code of length  over an alphabet with  symbols then, for every prime number . This condition is stronger than both the packing condition and the necessary condition given by the Lloyd theorem, as it for example excludes the existence of a perfect code with the parameters . [Copyright &y& Elsevier]

Abstract: The present paper is a continuation of the author’s work (Hachenberger (2001) ) on primitivity and complete normality. For certain 2-power extensions over a Galois field , we are going to establish the existence of a primitive element which simultaneously generates a normal basis over every intermediate field of . The main result is as follows: Let and let be the largest integer such that divides ; if , where , then there exists a primitive element in that is completely normal over . Our method not only shows existence but also gives a fairly large lower bound on the number of primitive completely normal elements. In the above case this number is at least . We are further going to discuss lower bounds on the number of such elements in -power extensions, where and , or where is an odd prime, or where is equal to the characteristic of the underlying field. [Copyright &y& Elsevier]

Abstract: J.A. Gallian has proved [J.A. Gallian, Labeling prisms and prism related graphs, Congr. Numer. 59 (1987) 89–100] that every cubic graph obtainable from a -cycle by adding its diameters (the so-called Moebius Ladder of order ) is graceful. Here, in the case of even, we propose a new graceful labeling that besides being simpler than Gallian’s one is able to give, at the same time, a graceful labeling of the prism of order . Most importantly in the case of odd, namely in the bipartite case, we prove that also admits an -labeling. This implies that there exists a cyclic decomposition of the complete graph into copies of for every pair of positive integers and with odd. In some cases we are able to give such decompositions also when is even. Apart from the case of that is an obvious consequence of the gracefulness of , this happens, for instance, when (mod 4) and is a prime. [Copyright &y& Elsevier]

Abstract: We determine the cyclic semi-regular subgroups of the 2-transitive permutation groups and with a suitable power of a prime number . [Copyright &y& Elsevier]

Abstract: Let be a finite abelian group and let be an integer. A sequence of elements in is called a -barycentric sequence if there exists such that . The -barycentric Davenport constant is defined to be the smallest number such that every sequence in of length contains a -barycentric subsequence. In this paper, we prove that if is a prime, then for , which improves a result of Delorme et al. [ABSTRACT FROM AUTHOR]

Abstract: A regular edge-transitive graph is said to be semisymmetric if it is not vertex-transitive. By Folkman [J. Folkman, Regular line-symmetric graphs, J. Combin Theory 3 (1967) 215–232], there is no semisymmetric graph of order or for a prime and by Malnič, et al. [A. Malnič, D. Marušič, C.Q. Wang, Cubic edge-transitive graphs of order , Discrete Math. 274 (2004) 187–198], there exists a unique cubic semisymmetric graph of order , the so-called Gray graph of order 54. In this paper it is shown that a connected cubic semisymmetric graph of order exists if and only if is divisible by 3. There are exactly two such graphs for a given order, which are constructed explicitly. [Copyright &y& Elsevier]

Abstract: A complete classification of -arc-transitive representations of primitive or bi-primitive groups of fourth-power-free order is given. The list consists of the following graphs: , , with an odd prime, Peterson graph , incidence and non-incidence graphs and of the Hadamard design on points, some explicit coset graphs of two-dimensional projective groups with and of Janko simple group , and the standard double cover of these coset graphs. Moreover, a classification of primitive permutation groups of fourth-power-free order is given. [Copyright &y& Elsevier]

Abstract: We examine the -ary codes, for any prime , that can be obtained from incidence matrices and line graphs of the Hamming graphs, , obtaining the main parameters of these codes. We show that the codes from the incidence matrices of can be used for full permutation decoding for all . [Copyright &y& Elsevier]

A map is called regular if its automorphism group acts regularly on the set of all flags (incident vertex–edge–face triples). An orientable map is called orientably regular if the group of all orientation-preserving automorphisms is regular on the set of all arcs (incident vertex–edge pairs). If an orientably regular map admits also orientation-reversing automorphisms, then it is regular, and is called reflexible. A regular embedding and orientably regular embedding of a graph are, respectively, 2-cell embeddings of as a regular map and orientably regular map on some closed surface. In Du et al. (2004) , the orientably regular embeddings of graphs of order for two primes and ( may be equal to ) have been classified, where all the reflexible maps can be easily read from the classification theorem. In , Du and Wang (2007) classified the nonorientable regular embeddings of these graphs for . In this paper, we shall classify the nonorientable regular embeddings of graphs of order where is a prime so that a complete classification of regular embeddings of graphs of order for two primes and is obtained. All graphs in this paper are connected and simple. [Copyright &y& Elsevier]