International Journal of Combinatorics http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. Modular Leech Trees of Order at Most 8 Thu, 18 Sep 2014 10:54:22 +0000 http://www.hindawi.com/journals/ijcom/2014/218086/ In 1975, John Leech asked when can the edges of a tree on vertices be labeled with positive integers such that the sums along the paths are exactly the integers . He found five such trees, and no additional trees have been discovered since. In 2011 Leach and Walsh introduced the idea of labeling trees with elements of the group where and examined the cases for . In this paper we show that no modular Leech trees of order 7 exist, and we find all modular Leech trees of order 8. David Leach Copyright © 2014 David Leach. All rights reserved. Midpoint-Free Subsets of the Real Numbers Tue, 26 Aug 2014 08:07:57 +0000 http://www.hindawi.com/journals/ijcom/2014/214637/ A set of reals is midpoint-free if it has no subset such that and . If and is midpoint-free, it is a maximal midpoint-free subset of if there is no midpoint-free set such that . In each of the cases , we determine two maximal midpoint-free subsets of characterised by digit constraints on the base 3 representations of their members. Roger B. Eggleton Copyright © 2014 Roger B. Eggleton. All rights reserved. Normal Edge-Transitive Cayley Graphs of the Group Tue, 19 Aug 2014 00:00:00 +0000 http://www.hindawi.com/journals/ijcom/2014/628214/ A Cayley graph of a group is called normal edge-transitive if the normalizer of the right representation of the group in the automorphism of the Cayley graph acts transitively on the set of edges of the graph. In this paper, we determine all connected normal edge-transitive Cayley graphs of the group . A. Assari and F. Sheikhmiri Copyright © 2014 A. Assari and F. Sheikhmiri. All rights reserved. On the Genus of the Zero-Divisor Graph of Tue, 22 Jul 2014 08:04:37 +0000 http://www.hindawi.com/journals/ijcom/2014/390732/ Let be a commutative ring with identity. The zero-divisor graph of , denoted , is the simple graph whose vertices are the nonzero zero-divisors of , and two distinct vertices and are linked by an edge if and only if . The genus of a simple graph is the smallest integer such that can be embedded into an orientable surface . In this paper, we determine that the genus of the zero-divisor graph of , the ring of integers modulo , is two or three. Huadong Su and Pailing Li Copyright © 2014 Huadong Su and Pailing Li. All rights reserved. A Weighted Regularity Lemma with Applications Thu, 19 Jun 2014 11:48:40 +0000 http://www.hindawi.com/journals/ijcom/2014/602657/ We prove an extension of the regularity lemma with vertex and edge weights which in principle can be applied for arbitrary graphs. The applications involve random graphs and a weighted version of the Erdős-Stone theorem. We also provide means to handle the otherwise uncontrolled exceptional set. Béla Csaba and András Pluhár Copyright © 2014 Béla Csaba and András Pluhár. All rights reserved. Decomposition Formulas for Triple -Hypergeometric Functions Thu, 15 May 2014 12:45:07 +0000 http://www.hindawi.com/journals/ijcom/2014/712321/ In the spirit of Hasanov, Srivastava, and Turaev (2006), we introduce new inverse operators together with a more general operator and find a summation formula for the last one. Based on these operators and the earlier known -analogues of the Burchnall-Chaundy operators, we find 15 symbolic operator formulas. Then, 10 expansions for the -analogues of Srivastava’s three triple hypergeometric functions in terms of -hypergeometric and -Kampé de Fériet functions are derived. These expansions readily reduce to 10 new expansions for the three triple Srivastava hypergeometric functions in terms of hypergeometric and Kampé de Fériet functions. Thomas Ernst Copyright © 2014 Thomas Ernst. All rights reserved. The Terminal Hosoya Polynomial of Some Families of Composite Graphs Wed, 16 Apr 2014 08:34:46 +0000 http://www.hindawi.com/journals/ijcom/2014/696507/ Let be a connected graph and let be the set of pendent vertices of . The terminal Hosoya polynomial of is defined as , where denotes the distance between the pendent vertices and . In this note paper we obtain closed formulae for the terminal Hosoya polynomial of rooted product graphs and corona product graphs. Emeric Deutsch and Juan Alberto Rodríguez-Velázquez Copyright © 2014 Emeric Deutsch and Juan Alberto Rodríguez-Velázquez. All rights reserved. Bounds on the Size of the Minimum Dominating Sets of Some Cylindrical Grid Graphs Mon, 07 Apr 2014 13:46:57 +0000 http://www.hindawi.com/journals/ijcom/2014/348359/ Let denote the domination number of the cylindrical grid graph formed by the Cartesian product of the graphs , the path of length m, and the graph , the cycle of length n, . In this paper we propose methods to find the domination numbers of graphs of the form with and and propose tight bounds on domination numbers of the graphs , . Moreover, we provide rough bounds on domination numbers of the graphs , and . We also point out how domination numbers and minimum dominating sets are useful for wireless sensor networks. Mrinal Nandi, Subrata Parui, and Avishek Adhikari Copyright © 2014 Mrinal Nandi et al. All rights reserved. On the Cardinality of the -Topologies on a Finite Set Mon, 31 Mar 2014 07:10:42 +0000 http://www.hindawi.com/journals/ijcom/2014/798074/ Let be the number of all labeled -topologies having open sets that we can define on points, and let be the number of those which are nonhomeomorphic. In this paper, we compute these numbers for and arbitrary . The numbers of all unlabeled and non--topologies with open sets are also given for . Messaoud Kolli Copyright © 2014 Messaoud Kolli. All rights reserved. Embedding Structures Associated with Riordan Arrays and Moment Matrices Mon, 17 Mar 2014 07:02:22 +0000 http://www.hindawi.com/journals/ijcom/2014/301394/ Every ordinary Riordan array contains two naturally embedded Riordan arrays. We explore this phenomenon, and we compare it to the situation for certain moment matrices of families of orthogonal polynomials. Paul Barry Copyright © 2014 Paul Barry. All rights reserved. The -Path Cover Polynomial of a Graph and a Model for General Coefficient Linear Recurrences Sun, 12 Jan 2014 00:00:00 +0000 http://www.hindawi.com/journals/ijcom/2014/258017/ An -path cover of a simple graph is a set of vertex disjoint paths of , each with vertices, that span . With every we associate a weight, , and define the weight of to be . The -path cover polynomial of is then defined as where the sum is taken over all -path covers of . This polynomial is a specialization of the path-cover polynomial of Farrell. We consider the -path cover polynomial of a weighted path and find the -term recurrence that it satisfies. The matrix form of this recurrence yields a formula equating the trace of the recurrence matrix with the -path cover polynomial of a suitably weighted cycle . A directed graph, , the edge-weighted -trellis, is introduced and so a third way to generate the solutions to the above -term recurrence is presented. We also give a model for general-term linear recurrences and time-dependent Markov chains. John P. McSorley and Philip Feinsilver Copyright © 2014 John P. McSorley and Philip Feinsilver. All rights reserved. On the Line Graph for Zero-Divisors of Tue, 31 Dec 2013 14:19:35 +0000 http://www.hindawi.com/journals/ijcom/2013/756179/ Let be a completely regular Hausdorff space and let be the ring of all continuous real valued functions defined on . In this paper, the line graph for the zero-divisor graph of is studied. It is shown that this graph is connected with diameter less than or equal to 3 and girth 3. It is shown that this graph is always triangulated and hypertriangulated. It is characterized when the graph is complemented. It is proved that the radius of this graph is 2 if and only if has isolated points; otherwise, the radius is 3. Bounds for the dominating number and clique number are also found in terms of the density number of . Ghada AlAfifi and Emad Abu Osba Copyright © 2013 Ghada AlAfifi and Emad Abu Osba. All rights reserved. The Linear 2- and 4-Arboricity of Complete Bipartite Graph Mon, 30 Dec 2013 11:21:31 +0000 http://www.hindawi.com/journals/ijcom/2013/501701/ A linear -forest of an undirected graph is a subgraph of whose components are paths with lengths at most . The linear -arboricity of , denoted by (), is the minimum number of linear -forests needed to decompose . In case the lengths of paths are not restricted, we then have the linear arboricity of , denoted by (). In this paper, the exact value of the linear 2- and 4-arboricity of complete bipartite graph for some and is obtained. Liancui Zuo, Bing Xue, and Shengjie He Copyright © 2013 Liancui Zuo et al. All rights reserved. On Cayley Digraphs That Do Not Have Hamiltonian Paths Thu, 26 Dec 2013 19:07:05 +0000 http://www.hindawi.com/journals/ijcom/2013/725809/ We construct an infinite family of connected, -generated Cayley digraphs that do not have hamiltonian paths, such that the orders of the generators and are unbounded. We also prove that if is any finite group with , then every connected Cayley digraph on has a hamiltonian path (but the conclusion does not always hold when or ). Dave Witte Morris Copyright © 2013 Dave Witte Morris. All rights reserved. Some New Results on Distance -Domination in Graphs Thu, 26 Dec 2013 11:07:22 +0000 http://www.hindawi.com/journals/ijcom/2013/795401/ We determine the distance -domination number for the total graph, shadow graph, and middle graph of path . Samir K. Vaidya and Nirang J. Kothari Copyright © 2013 Samir K. Vaidya and Nirang J. Kothari. All rights reserved. Some Inverse Relations Determined by Catalan Matrices Tue, 24 Sep 2013 11:17:23 +0000 http://www.hindawi.com/journals/ijcom/2013/528584/ We use the -sequence and -sequence of Riordan array to characterize the inverse relation associated with the Riordan array. We apply this result to prove some combinatorial identities involving Catalan matrices and binomial coefficients. Some matrix identities obtained by Shapiro and Radoux are all special cases of our identity. In addition, a unified form of Catalan matrices is introduced. Sheng-liang Yang Copyright © 2013 Sheng-liang Yang. All rights reserved. Gallai-Colorings of Triples and 2-Factors of Thu, 12 Sep 2013 11:30:27 +0000 http://www.hindawi.com/journals/ijcom/2013/929565/ A coloring of the edges of the -uniform complete hypergraph is a -coloring if there is no rainbow simplex; that is, every set of vertices contains two edges of the same color. The notion extends -colorings which are often called Gallai-colorings and originates from a seminal paper of Gallai. One well-known property of -colorings is that at least one color class has a spanning tree. J. Lehel and the senior author observed that this property does not hold for -colorings and proposed to study , the size of the largest monochromatic component which can be found in every -coloring of , the complete -uniform hypergraph. The previous remark says that and in this note, we address the case . We prove that and this determines for . We also prove that by excluding certain 2-factors from the middle layer of the Boolean lattice on seven elements. Lynn Chua, András Gyárfás, and Chetak Hossain Copyright © 2013 Lynn Chua et al. All rights reserved. Some New Classes of Open Distance-Pattern Uniform Graphs Wed, 24 Jul 2013 10:22:48 +0000 http://www.hindawi.com/journals/ijcom/2013/863439/ Given an arbitrary nonempty subset of vertices in a graph , each vertex in is associated with the set and called its open -distance-pattern. The graph is called open distance-pattern uniform (odpu-) graph if there exists a subset of such that for all and is called an open distance-pattern uniform (odpu-) set of The minimum cardinality of an odpu-set in , if it exists, is called the odpu-number of and is denoted by . Given some property , we establish characterization of odpu-graph with property . In this paper, we characterize odpu-chordal graphs, and thereby characterize interval graphs, split graphs, strongly chordal graphs, maximal outerplanar graphs, and ptolemaic graphs that are odpu-graphs. We also characterize odpu-self-complementary graphs, odpu-distance-hereditary graphs, and odpu-cographs. We prove that the odpu-number of cographs is even and establish that any graph can be embedded into a self-complementary odpu-graph , such that and are induced subgraphs of . We also prove that the odpu-number of a maximal outerplanar graph is either or . Bibin K. Jose Copyright © 2013 Bibin K. Jose. All rights reserved. On Bondage Numbers of Graphs: A Survey with Some Comments Mon, 13 May 2013 16:01:57 +0000 http://www.hindawi.com/journals/ijcom/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. Bounds for the Largest Laplacian Eigenvalue of Weighted Graphs Sun, 12 May 2013 09:07:25 +0000 http://www.hindawi.com/journals/ijcom/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. Finite 1-Regular Cayley Graphs of Valency 5 Wed, 27 Mar 2013 13:24:06 +0000 http://www.hindawi.com/journals/ijcom/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. Initial Ideals of Tangent Cones to the Richardson Varieties in the Orthogonal Grassmannian Tue, 26 Mar 2013 13:16:15 +0000 http://www.hindawi.com/journals/ijcom/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. Graphs with no Minor Containing a Fixed Edge Wed, 20 Mar 2013 08:41:38 +0000 http://www.hindawi.com/journals/ijcom/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. Sunlet Decomposition of Certain Equipartite Graphs Tue, 19 Mar 2013 08:54:00 +0000 http://www.hindawi.com/journals/ijcom/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. An Algebraic Representation of Graphs and Applications to Graph Enumeration Mon, 04 Mar 2013 10:04:41 +0000 http://www.hindawi.com/journals/ijcom/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. The Structure of Reduced Sudoku Grids and the Sudoku Symmetry Group Wed, 07 Nov 2012 07:40:08 +0000 http://www.hindawi.com/journals/ijcom/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. The Tutte Polynomial of Some Matroids Thu, 04 Oct 2012 09:50:27 +0000 http://www.hindawi.com/journals/ijcom/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. Combinatorial Proofs of Some Identities for Nonregular Continued Fractions Sat, 29 Sep 2012 04:31:22 +0000 http://www.hindawi.com/journals/ijcom/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. Graphs with Constant Sum of Domination and Inverse Domination Numbers Mon, 27 Aug 2012 14:21:08 +0000 http://www.hindawi.com/journals/ijcom/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. Algebraic Integers as Chromatic and Domination Roots Mon, 14 May 2012 09:34:23 +0000 http://www.hindawi.com/journals/ijcom/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.