http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2013 , Hindawi Publishing Corporation . All rights reserved. Mon, 13 May 2013 16:01:57 +0000 http://www.hindawi.com/journals/ijct/2013/595210/ The domination number of a graph is the smallest number of vertices which dominate all remaining vertices by edges of . The bondage number of a nonempty graph is the smallest number of edges whose removal from results in a graph with domination number greater than the domination number of . The concept of the bondage number was formally introduced by Fink et al. in 1990. Since then, this topic has received considerable research attention and made some progress, variations, and generalizations. This paper gives a survey on the bondage number, including known results, conjectures, problems, and some comments, also selectively summarizes other types of bondage numbers. Jun-Ming Xu Copyright © 2013 Jun-Ming Xu. All rights reserved. Sun, 12 May 2013 09:07:25 +0000 http://www.hindawi.com/journals/ijct/2013/520610/ Let be weighted graphs, as the graphs where the edge weights are positive definite matrices. The Laplacian eigenvalues of a graph are the eigenvalues of Laplacian matrix of a graph . We obtain two upper bounds for the largest Laplacian eigenvalue of weighted graphs and we compare these bounds with previously known bounds. Sezer Sorgun Copyright © 2013 Sezer Sorgun. All rights reserved. Wed, 27 Mar 2013 13:24:06 +0000 http://www.hindawi.com/journals/ijct/2013/125916/ Let and . We say is -regular Cayley graph if acts regularly on its arcs. is said to be core-free if is core-free in some . In this paper, we prove that if an -regular Cayley graph of valency is not normal or binormal, then it is the normal cover of one of two core-free ones up to isomorphism. In particular, there are no core-free -regular Cayley graphs of valency . Jing Jian Li, Ben Gong Lou, and Xiao Jun Zhang Copyright © 2013 Jing Jian Li et al. All rights reserved. Tue, 26 Mar 2013 13:16:15 +0000 http://www.hindawi.com/journals/ijct/2013/392437/ A Richardson variety in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety in the Orthogonal Grassmannian and an opposite Schubert variety therein. We give an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any T-fixed point of , thus generalizing a result of Raghavan and Upadhyay (2009). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the Orthogonal-bounded-RSK (OBRSK). Shyamashree Upadhyay Copyright © 2013 Shyamashree Upadhyay. All rights reserved. Wed, 20 Mar 2013 08:41:38 +0000 http://www.hindawi.com/journals/ijct/2013/783710/ It is well known that every cycle of a graph must intersect every cut in an even number of edges. For planar graphs, Ford and Fulkerson proved that, for any edge e, there exists a cycle containing e that intersects every minimal cut containing e in exactly two edges. The main result of this paper generalizes this result to any nonplanar graph G provided G does not have a minor containing the given edge e. Ford and Fulkerson used their result to provide an efficient algorithm for solving the maximum-flow problem on planar graphs. As a corollary to the main result of this paper, it is shown that the Ford-Fulkerson algorithm naturally extends to this more general class of graphs. Donald K. Wagner Copyright © 2013 Donald K. Wagner. All rights reserved. Tue, 19 Mar 2013 08:54:00 +0000 http://www.hindawi.com/journals/ijct/2013/907249/ Let stand for the sunlet graph which is a graph that consists of a cycle and an edge terminating in a vertex of degree one attached to each vertex of cycle . The necessary condition for the equipartite graph to be decomposed into for is that the order of must divide , the order of . In this work, we show that this condition is sufficient for the decomposition. The proofs are constructive using graph theory techniques. Abolape D. Akwu and Deborah O. A. Ajayi Copyright © 2013 Abolape D. Akwu and Deborah O. A. Ajayi. All rights reserved. Mon, 04 Mar 2013 10:04:41 +0000 http://www.hindawi.com/journals/ijct/2013/347613/ We give a recursion formula to generate all the equivalence classes of connected graphs with coefficients given by the inverses of the orders of their groups of automorphisms. We use an algebraic graph representation to apply the result to the enumeration of connected graphs, all of whose biconnected components have the same number of vertices and edges. The proof uses Abel’s binomial theorem and generalizes Dziobek’s induction proof of Cayley’s formula. Ângela Mestre Copyright © 2013 Ângela Mestre. All rights reserved. Wed, 07 Nov 2012 07:40:08 +0000 http://www.hindawi.com/journals/ijct/2012/760310/ A Sudoku grid is a constrained Latin square. In this paper a reduced Sudoku grid is described, the properties of which differ, through necessity, from that of a reduced Latin square. The Sudoku symmetry group is presented and applied to determine a mathematical relationship between the number of reduced Sudoku grids and the total number of Sudoku grids for any size. This relationship simplifies the enumeration of Sudoku grids and an example of the use of this method is given. Siân K. Jones, Stephanie Perkins, and Paul A. Roach Copyright © 2012 Siân K. Jones et al. All rights reserved. Thu, 04 Oct 2012 09:50:27 +0000 http://www.hindawi.com/journals/ijct/2012/430859/ The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any multiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of it. The deletion and contraction operations are natural reductions for many network models arising from a wide range of problems at the heart of computer science, engineering, optimization, physics, and biology. Even though the invariant is #P-hard to compute in general, there are many occasions when we face the task of computing the Tutte polynomial for some families of graphs or matroids. In this work, we compile known formulas for the Tutte polynomial of some families of graphs and matroids. Also, we give brief explanations of the techniques that were used to find the formulas. Hopefully, this will be useful for researchers in Combinatorics and elsewhere. Criel Merino, Marcelino Ramírez-Ibáñez, and Guadalupe Rodríguez-Sánchez Copyright © 2012 Criel Merino et al. All rights reserved. Sat, 29 Sep 2012 04:31:22 +0000 http://www.hindawi.com/journals/ijct/2012/894380/ A combinatorial interpretation of nonregular continued fractions is studied. Using a modification of a tiling technique due to Benjamin and Quinn, combinatorial proofs of some identities for nonregular continued fractions are obtained. Oranit Panprasitwech Copyright © 2012 Oranit Panprasitwech. All rights reserved. Mon, 27 Aug 2012 14:21:08 +0000 http://www.hindawi.com/journals/ijct/2012/831489/ A subset D of the vertex set of a graph G, is a dominating set if every vertex in 𝑉−𝐷 is adjacent to at least one vertex in D. The domination number 𝛾(𝐺) is the minimum cardinality of a dominating set of G. A subset of 𝑉−𝐷, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number 𝛾(𝐺) is the minimum cardinality of the inverse dominating sets. Domke et al. (2004) characterized connected graphs G with 𝛾(𝐺)+𝛾(𝐺)=𝑛, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs G with minimum degree at least two and 𝛾(𝐺)+𝛾(𝐺)=𝑛−1. T. Tamizh Chelvam and T. Asir Copyright © 2012 T. Tamizh Chelvam and T. Asir. All rights reserved. Mon, 14 May 2012 09:34:23 +0000 http://www.hindawi.com/journals/ijct/2012/780765/ Let 𝐺 be a simple graph of order 𝑛 and 𝜆∈ℕ. A mapping 𝑓∶𝑉(𝐺)→{1,2,…,𝜆} is called a 𝜆-colouring of 𝐺 if 𝑓(𝑢)≠𝑓(𝑣) whenever the vertices 𝑢 and 𝑣 are adjacent in 𝐺. The number of distinct 𝜆-colourings of 𝐺, denoted by 𝑃(𝐺,𝜆), is called the chromatic polynomial of 𝐺. The domination polynomial of 𝐺 is the polynomial ∑𝐷(𝐺,𝜆)=𝑛𝑖=1𝑑(𝐺,𝑖)𝜆𝑖, where 𝑑(𝐺,𝑖) is the number of dominating sets of 𝐺 of size 𝑖. Every root of 𝑃(𝐺,𝜆) and 𝐷(𝐺,𝜆) is called the chromatic root and the domination root of 𝐺, respectively. Since chromatic polynomial and domination polynomial are monic polynomial with integer coefficients, its zeros are algebraic integers. This naturally raises the question: which algebraic integers can occur as zeros of chromatic and domination polynomials? In this paper, we state some properties of this kind of algebraic integers. Saeid Alikhani and Roslan Hasni Copyright © 2012 Saeid Alikhani and Roslan Hasni. All rights reserved. Sun, 13 May 2012 11:12:17 +0000 http://www.hindawi.com/journals/ijct/2012/409505/ The generating function for a restricted partition function is derived. This in conjunction with two identities of Rogers provides new partition theoretic interpretations of Rogers-Ramanujan identities. A. K. Agarwal and M. Goyal Copyright © 2012 A. K. Agarwal and M. Goyal. All rights reserved. Tue, 07 Feb 2012 15:10:53 +0000 http://www.hindawi.com/journals/ijct/2012/908356/ In this work a convex relaxation of a subgraph isomorphism problem is proposed, which leads to a new lower bound that can provide a proof that a subgraph isomorphism between two graphs can not be found. The bound is based on a semidefinite programming relaxation of a combinatorial optimisation formulation for subgraph isomorphism and is explained in detail. We consider subgraph isomorphism problem instances of simple graphs which means that only the structural information of the two graphs is exploited and other information that might be available (e.g., node positions) is ignored. The bound is based on the fact that a subgraph isomorphism always leads to zero as lowest possible optimal objective value in the combinatorial problem formulation. Therefore, for problem instances with a lower bound that is larger than zero this represents a proof that a subgraph isomorphism can not exist. But note that conversely, a negative lower bound does not imply that a subgraph isomorphism must be present and only indicates that a subgraph isomorphism can not be excluded. In addition, the relation of our approach and the reformulation of the largest common subgraph problem into a maximum clique problem is discussed. Christian Schellewald Copyright © 2012 Christian Schellewald. All rights reserved. Mon, 06 Feb 2012 13:12:25 +0000 http://www.hindawi.com/journals/ijct/2012/406250/ We present two variations of the game 3-Euclid. The games involve a triplet of positive integers. Two players move alternately. In the first game, each move is to subtract a positive integer multiple of the smallest integer from one of the other integers as long as the result remains positive. In the second game, each move is to subtract a positive integer multiple of the smallest integer from the largest integer as long as the result remains positive. The player who makes the last move wins. We show that the two games have the same 𝒫-positions and positions of Sprague-Grundy value 1. We present three theorems on the periodicity of 𝒫-positions and positions of Sprague-Grundy value 1. We also obtain a theorem on the partition of Sprague-Grundy values for each game. In addition, we examine the misère versions of the two games and show that the Sprague-Grundy functions of each game and its misère version differ slightly. Nhan Bao Ho Copyright © 2012 Nhan Bao Ho. All rights reserved. Sun, 29 Jan 2012 08:11:04 +0000 http://www.hindawi.com/journals/ijct/2011/403140/ We tackle the combinatorics of coloured hard-dimer objects. This is achieved by identifying coloured hard-dimer configurations with a certain class of rooted trees that allow for an algebraic treatment in terms of noncommutative formal power series. A representation in terms of matrices then allows to find the asymptotic behaviour of these objects. Maria Simonetta Bernabei and Horst Thaler Copyright © 2011 Maria Simonetta Bernabei and Horst Thaler. All rights reserved. Fri, 27 Jan 2012 15:36:13 +0000 http://www.hindawi.com/journals/ijct/2012/957284/ Let Γ(ℤ𝑛[𝑖]) be the zero divisor graph for the ring of the Gaussian integers modulo 𝑛. Several properties of the line graph of Γ(ℤ𝑛[𝑖]), 𝐿(Γ(ℤ𝑛[𝑖])) are studied. It is determined when 𝐿(Γ(ℤ𝑛[𝑖])) is Eulerian, Hamiltonian, or planer. The girth, the diameter, the radius, and the chromatic and clique numbers of this graph are found. In addition, the domination number of 𝐿(Γ(ℤ𝑛[𝑖])) is given when 𝑛 is a power of a prime. On the other hand, several graph invariants for Γ(ℤ𝑛[𝑖]) are also determined. Khalida Nazzal and Manal Ghanem Copyright © 2012 Khalida Nazzal and Manal Ghanem. All rights reserved. Mon, 23 Jan 2012 11:41:32 +0000 http://www.hindawi.com/journals/ijct/2012/154281/ All extremal ternary self-dual codes of length 48 that have some automorphism of prime order 𝑝≥5 are equivalent to one of the two known codes, the Pless code or the extended quadratic residue code. Gabriele Nebe Copyright © 2012 Gabriele Nebe. All rights reserved. Thu, 29 Dec 2011 09:56:20 +0000 http://www.hindawi.com/journals/ijct/2011/824742/ We reexamine Albert and Nowakowski's variation on the game of Nim, called End-Nim, in which the players may only remove coins from the leftmost or rightmost piles. We reformulate Albert and Nowakowski's solution to this game. We examine its misère version and a further variant where the winner is the player who reduces the game to a single pile; we call this Loop-End-Nim. We show that the three games, End-Nim, misère-End-Nim, and Loop-End-Nim, all have the same losing positions, except for the positions where all the piles are of equal size. We also give some partial results concerning the higher Sprague-Grundy values of the three games. Grant Cairns and Nhan Bao Ho Copyright © 2011 Grant Cairns and Nhan Bao Ho. All rights reserved. Mon, 26 Dec 2011 15:11:44 +0000 http://www.hindawi.com/journals/ijct/2011/389369/ The 𝑘-tuple domination problem, for a fixed positive integer 𝑘, is to find a minimum size vertex subset such that every vertex in the graph is dominated by at least 𝑘 vertices in this set. The case when 𝑘=2 is called 2-tuple domination problem or double domination problem. In this paper, the 2-tuple domination problem is studied on interval graphs from an algorithmic point of view, which takes 𝑂(𝑛2) time, 𝑛 is the total number of vertices of the interval graph. Tarasankar Pramanik, Sukumar Mondal, and Madhumangal Pal Copyright © 2011 Tarasankar Pramanik et al. All rights reserved. Wed, 21 Dec 2011 11:30:32 +0000 http://www.hindawi.com/journals/ijct/2011/893061/ Fuzzy sets, rough sets, and later on IF sets became useful mathematical tools for solving various decision making problems and data mining problems. Molodtsov introduced another concept soft set theory as a general frame work for reasoning about vague concepts. Since most of the data collected are either linguistic variable or consist of vague concepts so IF set and soft set help a lot in data mining problem. The aim of this paper is to introduce the concept of IF soft lower rough approximation and IF upper rough set approximation. Also, some properties of this set are studied, and also some problems of decision making are cited where this concept may help. Further research will be needed to apply this concept fully in the decision making and data mining problems. Sharmistha Bhattacharya (Halder) and Bijan Davvaz Copyright © 2011 Sharmistha Bhattacharya (Halder) and Bijan Davvaz. All rights reserved. Wed, 07 Dec 2011 14:58:11 +0000 http://www.hindawi.com/journals/ijct/2012/909285/ The first and second Zagreb indices were first introduced by Gutman and Trinajstić (1972). It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical instances, and in elsewhere. Recently, the first and second Zagreb coindices, a new pair of invariants, were introduced in Došlić (2008). In this paper we introduce the 𝑎 and (𝑎,𝑏)-analogs of the above Zagreb indices and coindices and investigate the relationship between the enhanced versions to get a unified theory. Toufik Mansour and Chunwei Song Copyright © 2012 Toufik Mansour and Chunwei Song. All rights reserved. Tue, 11 Oct 2011 13:14:23 +0000 http://www.hindawi.com/journals/ijct/2011/101928/ For any 2-distance set 𝑋 in the n-dimensional binary Hamming space 𝐻𝑛, let Γ𝑋 be the graph with 𝑋 as the vertex set and with two vertices adjacent if and only if the distance between them is the smaller of the two nonzero distances in 𝑋. The binary spherical representation number of a graph Γ, or bsr(Γ), is the least n such that Γ is isomorphic to Γ𝑋, where 𝑋 is a 2-distance set lying on a sphere in 𝐻𝑛. It is shown that if Γ is a connected regular graph, then bsr(Γ)≥𝑏−𝑚, where b is the order of Γ and m is the multiplicity of the least eigenvalue of Γ, and the case of equality is characterized. In particular, if Γ is a connected strongly regular graph, then bsr(Γ)=𝑏−𝑚 if and only if Γ is the block graph of a quasisymmetric 2-design. It is also shown that if a connected regular graph is cospectral with a line graph and has the same binary spherical representation number as this line graph, then it is a line graph. Yury J. Ionin Copyright © 2011 Yury J. Ionin. All rights reserved. Sun, 04 Sep 2011 09:09:07 +0000 http://www.hindawi.com/journals/ijct/2011/137356/ We describe a construction of primitive 2-designs and strongly regular graphs from the simple groups 𝐿(3,5),𝑈(5,2), and 𝑆(6,2). The designs and the graphs are constructed by defining incidence structures on conjugacy classes of maximal subgroups of 𝐿(3,5),𝑈(5,2), and 𝑆(6,2). In addition, from the group 𝑆(6,2), we construct 2-designs with parameters (28,4,4) and (28,4,1) having the full automorphism group isomorphic to 𝑈(3,3)∶𝑍2. Dean Crnković and Vedrana Mikulić Crnković Copyright © 2011 Dean Crnković and Vedrana Mikulić Crnković. All rights reserved. Thu, 25 Aug 2011 13:04:45 +0000 http://www.hindawi.com/journals/ijct/2011/659567/ Recently, we identified a hierarchy relation between trinucleotide comma-free codes and trinucleotide circular codes (see our previous works). Here, we extend our hierarchy with two new classes of codes, called 𝐷𝐿𝐷 and 𝐿𝐷𝐿 codes, which are stronger than the comma-free codes. We also prove that no circular code with 20 trinucleotides is a 𝐷𝐿𝐷 code and that a circular code with 20 trinucleotides is comma-free if and only if it is a 𝐿𝐷𝐿 code. Finally, we point out the possible role of the symmetric group ∑4 in the mathematical study of trinucleotide circular codes. Christian J. Michel and Giuseppe Pirillo Copyright © 2011 Christian J. Michel and Giuseppe Pirillo. All rights reserved. Tue, 09 Aug 2011 08:33:02 +0000 http://www.hindawi.com/journals/ijct/2011/206930/ Suppose that 𝐺 is a finite group, such that |𝐺|=27𝑝, where 𝑝 is prime. We show that if 𝑆 is any generating set of 𝐺, then there is a Hamiltonian cycle in the corresponding Cayley graph Cay(𝐺;𝑆). Ebrahim Ghaderpour and Dave Witte Morris Copyright © 2011 Ebrahim Ghaderpour and Dave Witte Morris. All rights reserved. Wed, 03 Aug 2011 12:24:01 +0000 http://www.hindawi.com/journals/ijct/2012/273416/ It was proved (Bessy et al., 2010) that for 𝑟≥1, a tournament with minimum semidegree at least 2𝑟−1 contains at least r vertex-disjoint directed triangles. It was also proved (Lichiardopol, 2010) that for integers 𝑞≥3 and 𝑟≥1, every tournament with minimum semidegree at least (𝑞−1)𝑟−1 contains at least r vertex-disjoint directed cycles of length 𝑞. None information was given on these directed cycles. In this paper, we fill a little this gap. Namely, we prove that for 𝑑≥1 and 𝑟≥1, every tournament of minimum outdegree at least ((𝑑2+3𝑑+2)/2)𝑟−(𝑑2+𝑑+2)/2 contains at least 𝑟 vertex-disjoint strongly connected subtournaments of minimum outdegree 𝑑. Next, we prove for tournaments a conjecture of Stiebitz stating that for integers 𝑠≥1 and 𝑡≥1, there exists a least number 𝑓(𝑠,𝑡) such that every digraph with minimum outdegree at least 𝑓(𝑠,𝑡) can be vertex-partitioned into two sets inducing subdigraphs with minimum outdegree at least 𝑠 and at least 𝑡, respectively. Similar results related to the semidegree will be given. All these results are consequences of two results concerning the maximum order of a tournament of minimum outdegree 𝑑 (of minimum semidegree 𝑑) not containing proper subtournaments of minimum outdegree 𝑑 (of minimum semidegree 𝑑). Nicolas Lichiardopol Copyright © 2012 Nicolas Lichiardopol. All rights reserved. Wed, 03 Aug 2011 11:11:22 +0000 http://www.hindawi.com/journals/ijct/2012/284383/ A vertex irregular total 𝑘-labeling of a graph 𝐺 with vertex set 𝑉 and edge set 𝐸 is an assignment of positive integer labels {1,2,…,𝑘} to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of 𝐺, denoted by tvs(𝐺) is the minimum value of the largest label 𝑘 over all such irregular assignment. In this paper, we consider the total vertex irregularity strengths of disjoint union of 𝑠 isomorphic sun graphs, tvs(𝑠𝑀𝑛), disjoint union of 𝑠 consecutive nonisomorphic sun graphs, ⋃tvs(𝑠𝑖=1𝑀𝑖+2), and disjoint union of any two nonisomorphic sun graphs tvs(𝑀𝑘∪𝑀𝑛). Slamin, Dafik, and Wyse Winnona Copyright © 2012 Slamin et al. All rights reserved. Tue, 28 Jun 2011 14:14:24 +0000 http://www.hindawi.com/journals/ijct/2011/787596/ Expander graphs are widely used in communication problems and construction of error correcting codes. In such graphs, information gets through very quickly. Typically, it is not true for social or biological networks, though we may find a partition of the vertices such that the induced subgraphs on them and the bipartite subgraphs between any pair of them exhibit regular behavior of information flow within or between the vertex subsets. Implications between spectral and regularity properties are discussed. Marianna Bolla Copyright © 2011 Marianna Bolla. All rights reserved. Wed, 15 Jun 2011 16:16:40 +0000 http://www.hindawi.com/journals/ijct/2011/649687/ The graph Ramsey number 𝑅(𝐹1,𝐹2) is the smallest integer 𝑁 with the property that any complete graph of at least 𝑁 vertices whose edges are colored with two colors (say, red and blue) contains either a subgraph isomorphic to 𝐹1 all of whose edges are red or a subgraph isomorphic to 𝐹2 all of whose edges are blue. In this paper, we consider the Ramsey numbers for theta graphs. We determine 𝑅(𝜃4,𝜃𝑘), 𝑅(𝜃5,𝜃𝑘) for 𝑘≥4. More specifically, we establish that 𝑅(𝜃4,𝜃𝑘)=𝑅(𝜃5,𝜃𝑘)=2𝑘−1 for 𝑘≥7. Furthermore, we determine 𝑅(𝜃𝑛,𝜃𝑛) for 𝑛≥5. In fact, we establish that 𝑅(𝜃𝑛,𝜃𝑛)=(3𝑛/2)−1 if 𝑛 is even, 2𝑛−1 if 𝑛 is odd. M. M. M. Jaradat, M. S. A. Bataineh, and S. M. E. Radaideh Copyright © 2011 M. M. M. Jaradat et al. All rights reserved.