Contributions to Discrete Mathematics (E-Journal, University of Calgary)
Not a member yet
406 research outputs found
Sort by
Extended inverse theorems for -fold sumsets in integers
Full text linkLet , be integers and be a nonempty finite set of integers. Very recently, Tang and Xing studied extended inverse theorems for hk-h+1 < \left|hA\right| \leq hk+2h-3. In this paper, we extend the work of Tang and Xing and study all possible inverse theorems for hk-h+1<\left|hA\right| \leq hk+2h +1. Furthermore, we give a range of for which inverse problems are not possible
On some new families of -Mersenne and generalized -Gaussian Mersenne numbers and their polynomials
Full text linkIn this paper, we define the generalized k-Mersenne numbers and give a formula of generalized Mersenne polynomials and further, we study their properties. Moreover, we define Gaussian Mersenne numbers and obtain some identities like Binet Formula, Cassini\u27s identity, D\u27Ocagne\u27s Identity, and generating functions. The generalized Gaussian Mersenne numbers are described and their relation with the classical Mersenne numbers are explained. We also introduce a generalization of Gaussian Mersenne polynomials and establish some properties of these polynomials
Hankel Determinants of shifted sequences of Bernoulli and Euler numbers
Full text linkHankel determinants of sequences related to Bernoulli and Euler numbers have been studied before, and numerous identities are known. However, when a sequence is shifted by one unit, the situation often changes significantly. In this paper we use classical orthogonal polynomials and related methods to prove a general result concerning Hankel determinants for shifted sequences. We then apply this result to obtain new Hankel determinant evaluations for a total of 14 sequences related to Bernoulli and Euler numbers, one of which concerns Euler polynomials
Intriguing sets of strongly regular graphs and their related structures
Full text linkIn this paper we outline a technique for constructing directed strongly regular graphs by using strongly regular graphs having a "nice" family of intriguing sets. Further, we investigate such a construction method for rank three strongly regular graphs having at most vertices. Finally, several examples of intriguing sets of polar spaces are provided
On some partition theorems of M. V. Subbarao
Full text linkM.V. Subbarao proved that the number of partitions of in which parts occur with multiplicities 2, 3 and 5 is equal to the number of partitions of in which parts are congruent to , and generalized this result. In this paper, we give a new generalization of this identity and also present a new partition theorem in the spirit of Subbarao\u27s generalization of the identity
q-Analogues of -Series by Applying Carlitz Inversions to q-Pfaff-Saalschutz Theorem
Full text linkBy applying multiplicate forms of the Carlitz inverse series relations to the -Pfaff-Saalschtz summation theorem, we establish twenty five nonterminating -series identities with several of them serving as -analogues of infinite series expressions for and , including some typical ones discovered by Ramanujan (1914) and Guillera
Confining the robber on cographs
Full text linkIn a game of Cops and Robbers on graphs, usually the cops\u27 objective is to capture the robber---a situation which the robber wants to avoid invariably. In this paper, we begin with introducing the notions of trapping and confining the robber and discussing their relations with capturing the robber. Our goal is to study the confinement of the robber on graphs that are free of a fixed path as an induced subgraph. We present some necessary conditions for graphs not containing the path on vertices (referred to as -free graphs) for some , so that cops do not have a strategy to capture or confine the robber on (Propositions 2.1, 2.3). We then show that for planar cographs and planar -free graphs the confining cop number is at most one and two, respectively (Corollary 2.4). We also show that the number of vertices of a connected cograph on which one cop does not have a strategy to confine the robber has a tight lower bound of eight. Moreover, we explore the effects of twin operations---which are well known to provide a characterization of cographs---on the number of cops required to capture or confine the robber on cographs. Finally, we pose two conjectures on confining the robber on -free graphs and the smallest planar graph of confining cop number of three
On a generalized basic series and Rogers–Ramanujan type identities - II
Full text linkThis paper is in continuation with our recent paper “On a generalized basic series and Rogers-Ramanujan type identities”. Here, we consider two generalized basic series and interpret these basic series as the generating function of some restricted -color partitions and restricted weighted lattice paths. The basic series discussed in the aforementioned paper, is now a mere particular case of one of the generalized basic series that are discussed in this paper. Besides, eight particular cases are also discussed which give combinatorial interpretations of eight Rogers–Ramanujan type identities which are combinatorially unexplored till date
A graph related to the sum of element orders of a finite group
Full text linkA finite group is called -divisible iff for any subgroup of a finite group . Here, is the sum of element orders of . For now, the only known examples of such groups are the cyclic ones of square-free order. The existence of non-abelian -divisible groups still constitutes an open question. The aim of this paper is to make a connection between the -divisibility property and graph theory. Hence, for a finite group , we introduce a simple undirected graph called the -divisibility graph of . We denote it by . Its vertices are the non-trivial subgroups of , while two distinct vertices and are adjacent iff and or and . We prove that is -divisible iff has a universal (dominating) vertex. Also, we study various properties of , when is a finite cyclic group. The choice of restricting our study to this specific class of groups is motivated in the paper
On Colouring Oriented Graphs of Large Girth
Full text linkWe prove that for every oriented graph and every choice of positive integers and , there exists an oriented graph along with a surjective homomorphism such that: (i) girth; (ii) for every oriented graph with at most vertices, there exists a homomorphism from to if and only if there exists a homomorphism from to C; and (iii) for every -pointed oriented graph with at most vertices and for every homomorphism there exists a unique homomorphism such that . Determining the chromatic number of an oriented graph is equivalent to finding the smallest integer such that admits a homomorphism to an order- tournament, so our main theorem yields results on the girth and chromatic number of oriented graphs. While our main proof is probabilistic (hence nonconstructive), for any given and , we include a construction of an oriented graph with girth and chromatic number