ISRN Combinatorics
http://www.hindawi.com
The latest articles from Hindawi Publishing Corporation
© 2014 , Hindawi Publishing Corporation . All rights reserved.

On Normal Ary Codes in RosenbloomTsfasman Metric
Wed, 02 Apr 2014 00:00:00 +0000
http://www.hindawi.com/journals/isrn.combinatorics/2014/237915/
The notion of normality of codes in Hamming metric is extended to the codes in RosenbloomTsfasman metric (RTmetric, in short). Using concepts of partition number and cell of codes in RTmetric, 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 RTmetric.
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
http://www.hindawi.com/journals/isrn.combinatorics/2014/316535/
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
http://www.hindawi.com/journals/isrn.combinatorics/2014/607018/
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
http://www.hindawi.com/journals/isrn.combinatorics/2014/897834/
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 DumontThomas Numeration Systems
Thu, 23 Jan 2014 08:15:48 +0000
http://www.hindawi.com/journals/isrn.combinatorics/2014/275260/
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 DumontThomas 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
http://www.hindawi.com/journals/isrn.combinatorics/2013/984549/
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 4Cycle with a Pendant Edge
Thu, 26 Sep 2013 09:32:34 +0000
http://www.hindawi.com/journals/isrn.combinatorics/2013/236298/
We consider the packings and coverings of complete graphs with isomorphic copies of the 4cycle 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
http://www.hindawi.com/journals/isrn.combinatorics/2013/792743/
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
http://www.hindawi.com/journals/isrn.combinatorics/2013/243981/
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
http://www.hindawi.com/journals/isrn.combinatorics/2013/398641/
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
http://www.hindawi.com/journals/isrn.combinatorics/2013/605279/
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
http://www.hindawi.com/journals/isrn.combinatorics/2013/703989/
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
http://www.hindawi.com/journals/isrn.combinatorics/2013/146595/
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
http://www.hindawi.com/journals/isrn.combinatorics/2013/201654/
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
http://www.hindawi.com/journals/isrn.combinatorics/2013/431024/
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
http://www.hindawi.com/journals/isrn.combinatorics/2013/759641/
We define the convolved kFibonacci numbers as an extension of the classical convolved Fibonacci numbers. Then we give some combinatorial formulas involving the kFibonacci and kLucas numbers. Moreover we obtain the convolved kFibonacci 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
http://www.hindawi.com/journals/isrn.combinatorics/2013/398473/
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.
HungChih Lee and YenPo Chu
Copyright © 2013 HungChih Lee and YenPo Chu. All rights reserved.

Disconnected Forbidden Subgraphs, Toughness and Hamilton Cycles
Sun, 10 Mar 2013 15:58:29 +0000
http://www.hindawi.com/journals/isrn.combinatorics/2013/673971/
In 1974, Goodman and Hedetniemi proved that every 2connected 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 2connected 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 1tough free graph is hamiltonian. We conjecture that every 1tough free graph is hamiltonian and every 1tough 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
http://www.hindawi.com/journals/isrn.combinatorics/2013/906756/
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.

ShatteringExtremal Set Systems of Small VCDimension
Wed, 27 Feb 2013 14:52:12 +0000
http://www.hindawi.com/journals/isrn.combinatorics/2013/126214/
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 shatteringextremal if it shatters exactly sets. We characterize shattering extremal set systems of VapnikChervonenkis dimension 1 in terms of their inclusion graphs. Also, from the perspective of extremality, we relate set systems of bounded VapnikChervonenkis 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
http://www.hindawi.com/journals/isrn.combinatorics/2013/724049/
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
http://www.hindawi.com/journals/isrn.combinatorics/2013/354696/
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 PatternMatching Conditions for
Wed, 13 Feb 2013 11:12:38 +0000
http://www.hindawi.com/journals/isrn.combinatorics/2013/634823/
We derive several multivariable generating functions for a generalized patternmatching 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
http://www.hindawi.com/journals/isrn.combinatorics/2013/672731/
Let 4HPMD denote a holey perfect Mendelsohn design with block size four. The existence of 4HPMDs 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 4HPMDs of type for : a 4HPMD() exists if and only if , except possibly for , (7, 6), (11, 9), (11, 10). We also investigate the existence of 4HPMD() for general and prove that there exists a 4HPMD() for all . Moreover, if , then a 4HPMD() exists for all ; if , then a 4HPMD() 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
http://www.hindawi.com/journals/isrn.combinatorics/2013/436987/
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
http://www.hindawi.com/journals/isrn.combinatorics/2013/479408/
Frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies. Frequency hopping sequences are needed in FHCDMA systems. In this paper, a construction of optimal sets of frequency hopping sequences is presented. The construction is based on the settheoretic 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
http://www.hindawi.com/journals/isrn.combinatorics/2013/806583/
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
http://www.hindawi.com/journals/isrn.combinatorics/2013/358527/
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
http://www.hindawi.com/journals/isrn.combinatorics/2013/414623/
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
http://www.hindawi.com/journals/isrn.combinatorics/2013/795427/
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.