ISRN Combinatorics The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. On Normal -Ary Codes in Rosenbloom-Tsfasman Metric Wed, 02 Apr 2014 00:00:00 +0000 The notion of normality of codes in Hamming metric is extended to the codes in Rosenbloom-Tsfasman metric (RT-metric, in short). Using concepts of partition number and -cell of codes in RT-metric, we establish results on covering radius and normality of -ary codes in this metric. We also examine the acceptability of various coordinate positions of -ary codes in this metric. And thus, by exploring the feasibility of applying amalgamated direct sum method for construction of codes, we analyze the significance of normality in RT-metric. R. S. Selvaraj and Venkatrajam Marka Copyright © 2014 R. S. Selvaraj and Venkatrajam Marka. All rights reserved. Noncontiguous Pattern Containment in Binary Trees Mon, 24 Mar 2014 14:35:02 +0000 We consider the enumeration of binary trees containing noncontiguous binary tree patterns. First, we show that any two -leaf binary trees are contained in the set of all -leaf trees the same number of times. We give a functional equation for the multivariate generating function for number of -leaf trees containing a specified number of copies of any path tree, and we analyze tree patterns with at most 4 leaves. The paper concludes with implications for pattern containment in permutations. Lara Pudwell, Connor Scholten, Tyler Schrock, and Alexa Serrato Copyright © 2014 Lara Pudwell et al. All rights reserved. Some New Results on Prime Cordial Labeling Sun, 23 Mar 2014 14:45:30 +0000 A prime cordial labeling of a graph with the vertex set is a bijection such that each edge is assigned the label 1 if and 0 if ; then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph which admits a prime cordial labeling is called a prime cordial graph. In this work we give a method to construct larger prime cordial graph using a given prime cordial graph . In addition to this we have investigated the prime cordial labeling for double fan and degree splitting graphs of path as well as bistar. Moreover we prove that the graph obtained by duplication of an edge (spoke as well as rim) in wheel admits prime cordial labeling. S. K. Vaidya and N. H. Shah Copyright © 2014 S. K. Vaidya and N. H. Shah. All rights reserved. The Diophantine Equation and -Balancing Numbers Thu, 20 Mar 2014 12:24:01 +0000 Let be an odd integer such that is a prime. In this work, we determine all integer solutions of the Diophantine equation and then we deduce the general terms of all -balancing numbers. Ahmet Tekcan, Merve Tayat, and Meltem E. Özbek Copyright © 2014 Ahmet Tekcan et al. All rights reserved. Integer Semigroups Associated with Dumont-Thomas Numeration Systems Thu, 23 Jan 2014 08:15:48 +0000 Given a primitive substitution, we define different binary operations on infinite subsets of the nonnegative integers. These binary operations are defined with the help of the Dumont-Thomas numeration system; that is, a numeration system associated with the substitution. We give conditions for these semigroups to have an identity element. We show that they are not finitely generated. These semigroups define actions on the set of positive integers. We describe the orbits of these actions. We also estimate the density of these sets as subsets of the positive integers. Víctor F. Sirvent Copyright © 2014 Víctor F. Sirvent. All rights reserved. -Tuple Total Restrained Domination in Complementary Prisms Sun, 17 Nov 2013 15:08:31 +0000 In a graph with , a -tuple total restrained dominating set is a subset of such that each vertex of   is adjacent to at least vertices of and also each vertex of is adjacent to at least vertices of  . The minimum number of vertices of such sets in is the -tuple total restrained domination number of . In [-tuple total restrained domination/domatic in graphs, BIMS], the author initiated the study of the -tuple total restrained domination number in graphs. In this paper, we continue it in the complementary prism of a graph. Adel P. Kazemi Copyright © 2013 Adel P. Kazemi. All rights reserved. Packings and Coverings of Various Complete Graphs with the 4-Cycle with a Pendant Edge Thu, 26 Sep 2013 09:32:34 +0000 We consider the packings and coverings of complete graphs with isomorphic copies of the 4-cycle with a pendant edge. Necessary and sufficient conditions are given for such structures for (1) complete graphs , (2) complete bipartite graphs , and (3) complete graphs with a hole . In the last two cases, we address both restricted and unrestricted coverings. Brandon Coker, Gary D. Coker, Robert Gardner, and Yan Xia Copyright © 2013 Brandon Coker et al. All rights reserved. Extended Chain of Domination Parameters in Graphs Thu, 26 Sep 2013 09:31:40 +0000 A subset of the vertex set of a graph is called an isolate set if the subgraph induced by has an isolated vertex. The subset is called an isolate dominating set if it is both isolate and dominating. Also, is called an isolate irredundant set if it is both isolate and irredundant. In this paper, we establish a chain connecting various isolate parameters with the existing domination parameters and discuss equality among the parameters in the extended chain. I. Sahul Hamid and S. Balamurugan Copyright © 2013 I. Sahul Hamid and S. Balamurugan. All rights reserved. Lexicographic Product and Isoperimetric Number Mon, 02 Sep 2013 13:16:37 +0000 The isoperimetric number of a graph , denoted by , was introduced by Mohar (1987). A graph and a subset of its vertices are given, and let denote the edge boundary of , the set of edges which connects vertices in to vertices not in . The isoperimetric number of is defined as . In this paper, some results about the isoperimetric number of graphs obtained by graph operations are given. Ersin Aslan and Alpay Kirlangic Copyright © 2013 Ersin Aslan and Alpay Kirlangic. All rights reserved. Isomorphic Digraphs from Affine Maps of Finite Cyclic Groups Mon, 02 Sep 2013 11:17:02 +0000 Let be a positive integer. For any pair of integers and , let be the digraph whose set of vertices is , and there exists a directed edge from vertex to vertex if . In this paper, we obtain a necessary and sufficient condition for which . Guixin Deng Copyright © 2013 Guixin Deng. All rights reserved. Freely Solvable Graphs in Peg Solitaire Mon, 02 Sep 2013 09:10:03 +0000 In a 2011 paper, the game of peg solitaire is generalized to arbitrary boards, which are treated as graphs in the combinatorial sense. Of particular interest are graphs that are freely solvable, that is, graphs that can be solved from any starting position. In this paper we give several examples of freely solvable graphs including all such trees with ten vertices or less, numerous cycles with a subdivided chord, meshes, and generalizations of the wheel, helm, and web. Robert A. Beeler and Aaron D. Gray Copyright © 2013 Robert A. Beeler and Aaron D. Gray. All rights reserved. A Polynomial Representation and a Unique Code of a Simple Undirected Graph Tue, 27 Aug 2013 14:52:34 +0000 We introduce a representation of simple undirected graphs in terms of polynomials and obtain a unique code for a simple undirected graph. Shamik Ghosh, Raibatak Sen Gupta, and M. K. Sen Copyright © 2013 Shamik Ghosh et al. All rights reserved. On the Domination Polynomial of Some Graph Operations Tue, 27 Aug 2013 09:44:48 +0000 Let be a simple graph of order . The domination polynomial of is the polynomial , where is the number of dominating sets of of size . Every root of is called the domination root of . In this paper, we study the domination polynomial of some graph operations. Saeid Alikhani Copyright © 2013 Saeid Alikhani. All rights reserved. Some New Perspectives on Global Domination in Graphs Tue, 20 Aug 2013 10:12:54 +0000 A dominating set is called a global dominating set if it is a dominating set of a graph and its complement . Here we explore the possibility to relate the domination number of graph and the global domination number of the larger graph obtained from by means of various graph operations. In this paper we consider the following problem: Does the global domination number remain invariant under any graph operations? We present an affirmative answer to this problem and establish several results. S. K. Vaidya and R. M. Pandit Copyright © 2013 S. K. Vaidya and R. M. Pandit. All rights reserved. Crystal Bases as Tuples of Integer Sequences Sun, 07 Jul 2013 15:58:48 +0000 We describe a set consisting of tuples of integer sequences and provide certain explicit maps on it. We show that this defines a semiregular crystal for and , respectively. Furthermore, we define for any dominant integral weight a connected subcrystal in , such that this crystal is isomorphic to the crystal graph . Finally, we provide an explicit description of these connected crystals . Deniz Kus Copyright © 2013 Deniz Kus. All rights reserved. Some Properties of Convolved -Fibonacci Numbers Wed, 12 Jun 2013 13:46:05 +0000 We define the convolved k-Fibonacci numbers as an extension of the classical convolved Fibonacci numbers. Then we give some combinatorial formulas involving the k-Fibonacci and k-Lucas numbers. Moreover we obtain the convolved k-Fibonacci numbers from a family of Hessenberg matrices. José L. Ramírez Copyright © 2013 José L. Ramírez. All rights reserved. Multidecompositions of the Balanced Complete Bipartite Graph into Paths and Stars Mon, 11 Mar 2013 13:07:00 +0000 Let and denote a path and a star with edges, respectively. For graphs , , and , a -multidecomposition of is a partition of the edge set of into copies of and copies of with at least one copy of and at least one copy of . In this paper, necessary and sufficient conditions for the existence of the (, )-multidecomposition of the balanced complete bipartite graph are given. Hung-Chih Lee and Yen-Po Chu Copyright © 2013 Hung-Chih Lee and Yen-Po Chu. All rights reserved. Disconnected Forbidden Subgraphs, Toughness and Hamilton Cycles Sun, 10 Mar 2013 15:58:29 +0000 In 1974, Goodman and Hedetniemi proved that every 2-connected -free graph is hamiltonian. This result gave rise many other conditions for Hamilton cycles concerning various pairs and triples of forbidden connected subgraphs under additional connectivity conditions. In this paper we investigate analogous problems when forbidden subgraphs are disconnected which affects more global structures in graphs such as tough structures instead of traditional connectivity structures. In 1997, it was proved that a single forbidden connected subgraph in 2-connected graphs can create only a trivial class of hamiltonian graphs (complete graphs) with . In this paper we prove that a single forbidden subgraph can create a non trivial class of hamiltonian graphs if is disconnected: every -free graph either is hamiltonian or belongs to a well defined class of non hamiltonian graphs; every 1-tough -free graph is hamiltonian. We conjecture that every 1-tough -free graph is hamiltonian and every 1-tough -free graph is hamiltonian. Zh. G. Nikoghosyan Copyright © 2013 Zh. G. Nikoghosyan. All rights reserved. Wiener Index of Graphs with Radius Two Sun, 03 Mar 2013 14:13:40 +0000 The Wiener index of a graph is the sum of the distances between all pairs of vertices. It has been one of main descriptors that correlate a chemical compound's molecular graph with experimentally gathered data regarding the compound's characteristics. We characterize graphs with the maximum Wiener index among all graphs of order . with radius two. In addition, we pose a conjecture concerning the minimum Wiener index of graphs with given radius. If this conjecture is true, it will be able to answer an open question by You and Liu (2011). Yin Chen, Baoyindureng Wu, and Xinhui An Copyright © 2013 Yin Chen et al. All rights reserved. Shattering-Extremal Set Systems of Small VC-Dimension Wed, 27 Feb 2013 14:52:12 +0000 We say that a set system shatters a given set if . The Sauer inequality states that in general, a set system shatters at least sets. Here, we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly sets. We characterize shattering extremal set systems of Vapnik-Chervonenkis dimension 1 in terms of their inclusion graphs. Also, from the perspective of extremality, we relate set systems of bounded Vapnik-Chervonenkis dimension to their projections. Tamás Mészáros and Lajos Rónyai Copyright © 2013 Tamás Mészáros and Lajos Rónyai. All rights reserved. On Metric Dimension of Some Rotationally Symmetric Graphs Sun, 24 Feb 2013 11:39:10 +0000 A family of connected graphs is a family with constant metric dimension if dim() is finite and does not depend upon the choice of in . In this paper, we show that the graph and the graph obtained from the antiprism graph have constant metric dimension. M. Ali, M. T. Rahim, G. Ali, and U. Ali Copyright © 2013 M. Ali et al. All rights reserved. On Graphs Related to Comaximal Ideals of a Commutative Ring Tue, 19 Feb 2013 18:49:23 +0000 We study the co maximal graph , the induced subgraph of whose vertex set is , and a retract of , where is a commutative ring. For a graph which contains a cycle, we show that the core of is a union of triangles and rectangles, while a vertex in is either an end vertex or a vertex in the core. For a nonlocal ring , we prove that both the chromatic number and clique number of are identical with the number of maximal ideals of . A graph is also introduced on the vertex set , and graph properties of are studied. Tongsuo Wu, Meng Ye, Dancheng Lu, and Houyi Yu Copyright © 2013 Tongsuo Wu et al. All rights reserved. Generalized Pattern-Matching Conditions for Wed, 13 Feb 2013 11:12:38 +0000 We derive several multivariable generating functions for a generalized pattern-matching condition on the wreath product of the cyclic group and the symmetric group . In particular, we derive the generating functions for the number of matches that occur in elements of for any pattern of length 2 by applying appropriate homomorphisms from the ring of symmetric functions over an infinite number of variables to simple symmetric function identities. This allows us to derive several natural analogues of the distribution of rises relative to the product order on elements of . Our research leads to connections to many known objects/structures yet to be explained combinatorially. Sergey Kitaev, Andrew Niedermaier, Jeffrey Remmel, and Manda Riehl Copyright © 2013 Sergey Kitaev et al. All rights reserved. Holey Perfect Mendelsohn Designs of Type with Block Size Four Wed, 06 Feb 2013 14:38:58 +0000 Let 4-HPMD denote a holey perfect Mendelsohn design with block size four. The existence of 4-HPMDs with holes of size 2 and one hole of size 3, that is, of type , was established by Bennett et al. in 1997. In this paper, we investigate the existence of 4-HPMDs of type for : a 4-HPMD() exists if and only if , except possibly for , (7, 6), (11, 9), (11, 10). We also investigate the existence of 4-HPMD() for general and prove that there exists a 4-HPMD() for all . Moreover, if , then a 4-HPMD() exists for all ; if , then a 4-HPMD() exists for all . Hantao Zhang Copyright © 2013 Hantao Zhang. All rights reserved. The - Designs with Unsolvable Block Transitive Automorphism Wed, 30 Jan 2013 16:28:29 +0000 This paper is a contribution to the study of the automorphism groups of - designs. Let be - design and Aut a block transitive and a point primitive. If is unsolvable, then Soc, the socle of , is not . Kun Zhao and Shaojun Dai Copyright © 2013 Kun Zhao and Shaojun Dai. All rights reserved. Construction of Optimal Sets of Frequency Hopping Sequences Wed, 30 Jan 2013 13:23:56 +0000 Frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies. Frequency hopping sequences are needed in FH-CDMA systems. In this paper, a construction of optimal sets of frequency hopping sequences is presented. The construction is based on the set-theoretic characterization of an optimal set of FH sequences. Bin Wen Copyright © 2013 Bin Wen. All rights reserved. Generalized Pattern Avoidance Condition for the Wreath Product of Cyclic Groups with Symmetric Groups Thu, 17 Jan 2013 14:36:14 +0000 We continue the study of the generalized pattern avoidance condition for , the wreath product of the cyclic group with the symmetric group , initiated in the work by Kitaev et al., In press. Among our results, there are a number of (multivariable) generating functions both for consecutive and nonconsecutive patterns, as well as a bijective proof for a new sequence counted by the Catalan numbers. Sergey Kitaev, Jeffrey Remmel, and Manda Riehl Copyright © 2013 Sergey Kitaev et al. All rights reserved. Graphs Whose Edge Set Can Be Partitioned into Maximum Matchings Thu, 17 Jan 2013 13:58:40 +0000 This paper provides structural characterization of simple graphs whose edge set can be partitioned into maximum matchings. We use Vizing's classification of simple graphs based on edge chromatic index. Niraj Khare Copyright © 2013 Niraj Khare. All rights reserved. Closed Form Continued Fraction Expansions of Special Quadratic Irrationals Thu, 17 Jan 2013 13:39:08 +0000 We derive closed form expressions for the continued fractions of powers of certain quadratic surds. Specifically, consider the recurrence relation with , , a positive integer, and (note that gives the Fibonacci numbers). Let . We find simple closed form continued fraction expansions for for any integer by exploiting elementary properties of the recurrence relation and continued fractions. Daniel Fishman and Steven J. Miller Copyright © 2013 Daniel Fishman and Steven J. Miller. All rights reserved. Domination Integrity of Splitting Graph of Path and Cycle Thu, 17 Jan 2013 09:35:21 +0000 If is a dominating set of a connected graph then the domination integrity is the minimum of the sum of two parameters, the number of elements in and the order of the maximum component of . We investigate domination integrity of splitting graph of path and cycle . This work is an effort to relate network expansion and vulnerability parameter. Samir K. Vaidya and Nirang J. Kothari Copyright © 2013 Samir K. Vaidya and Nirang J. Kothari. All rights reserved.