http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2013 , Hindawi Publishing Corporation . All rights reserved. Mon, 11 Mar 2013 13:07:00 +0000 http://www.hindawi.com/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. Hung-Chih Lee and Yen-Po Chu Copyright © 2013 Hung-Chih Lee and Yen-Po Chu. All rights reserved. Sun, 10 Mar 2013 15:58:29 +0000 http://www.hindawi.com/isrn/combinatorics/2013/673971/ 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. Sun, 03 Mar 2013 14:13:40 +0000 http://www.hindawi.com/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. Wed, 27 Feb 2013 14:52:12 +0000 http://www.hindawi.com/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 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. Sun, 24 Feb 2013 11:39:10 +0000 http://www.hindawi.com/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. Tue, 19 Feb 2013 18:49:23 +0000 http://www.hindawi.com/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. Wed, 13 Feb 2013 11:12:38 +0000 http://www.hindawi.com/isrn/combinatorics/2013/634823/ 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. Wed, 06 Feb 2013 14:38:58 +0000 http://www.hindawi.com/isrn/combinatorics/2013/672731/ 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. Wed, 30 Jan 2013 16:28:29 +0000 http://www.hindawi.com/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. Wed, 30 Jan 2013 13:23:56 +0000 http://www.hindawi.com/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 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. Thu, 17 Jan 2013 14:36:14 +0000 http://www.hindawi.com/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. Thu, 17 Jan 2013 13:58:40 +0000 http://www.hindawi.com/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. Thu, 17 Jan 2013 13:39:08 +0000 http://www.hindawi.com/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. Thu, 17 Jan 2013 09:35:21 +0000 http://www.hindawi.com/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. Sun, 13 Jan 2013 15:48:56 +0000 http://www.hindawi.com/isrn/combinatorics/2013/453808/ Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs). Francesco M. Malvestuto Copyright © 2013 Francesco M. Malvestuto. All rights reserved. Sun, 13 Jan 2013 08:33:05 +0000 http://www.hindawi.com/isrn/combinatorics/2013/326038/ A radio labeling of a graph is a function from the vertex set to the set of nonnegative integers such that , where and are diameter and distance between and in graph , respectively. The radio number of is the smallest number such that has radio labeling with . We investigate radio number for total graph of paths. S. K. Vaidya and D. D. Bantva Copyright © 2013 S. K. Vaidya and D. D. Bantva. All rights reserved. Sun, 13 Jan 2013 07:58:27 +0000 http://www.hindawi.com/isrn/combinatorics/2013/107454/ We define a map between the symmetric group and the set of pairs of Dyck paths of semilength . We show that the map is injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map. Marilena Barnabei, Flavio Bonetti, and Matteo Silimbani Copyright © 2013 Marilena Barnabei et al. All rights reserved. Sun, 13 Jan 2013 07:55:33 +0000 http://www.hindawi.com/isrn/combinatorics/2013/615703/ Let be nonnegative integers. In this paper we study the basic properties of a discrete dynamical model of signed integer partitions that we denote by . A generic element of this model is a signed integer partition with exactly all distinct nonzero parts, whose maximum positive summand is not exceeding and whose minimum negative summand is not less than . In particular, we determine the covering relations, the rank function, and the parallel convergence time from the bottom to the top of by using an abstract Sand Piles Model with three evolution rules. The lattice was introduced by the first two authors in order to study some combinatorial extremal sum problems. C. Bisi, G. Chiaselotti, and P. A. Oliverio Copyright © 2013 C. Bisi et al. All rights reserved. Sun, 13 Jan 2013 07:51:07 +0000 http://www.hindawi.com/isrn/combinatorics/2013/574578/ The two operations, deletion and contraction of an edge, on multigraphs directly lead to the Tutte polynomial which satisfies a universal problem. As observed by Brylawski (1972) in terms of order relations, these operations may be interpreted as a particular instance of a general theory which involves universal invariants like the Tutte polynomial and a universal group, called the Tutte-Grothendieck group. In this contribution, Brylawski’s theory is extended in two ways: first of all, the order relation is replaced by a string rewriting system, and secondly, commutativity by partial commutations (that permits a kind of interpolation between noncommutativity and full commutativity). This allows us to clarify the relations between the semigroup subject to rewriting and the Tutte-Grothendieck group: the latter is actually the Grothendieck group completion of the former, up to the free adjunction of a unit (this was not even mentioned by Brylawski), and normal forms may be seen as universal invariants. Moreover we prove that such universal constructions are also possible in case of a nonconvergent rewriting system, outside the scope of Brylawski’s work. Laurent Poinsot Copyright © 2013 Laurent Poinsot. All rights reserved. Tue, 25 Dec 2012 15:14:21 +0000 http://www.hindawi.com/isrn/combinatorics/2013/363724/ For a positive integer let be the th harmonic number. In this paper we prove that, for any prime ,  . Notice that the first part of this congruence is proposed in 2008 by Tauraso. In our elementary proof of the second part of the above congruence we use certain classical congruences modulo a prime and the square of a prime, some congruences involving harmonic numbers, and a combinatorial identity due to Hernández. Our auxiliary results contain many interesting combinatorial congruences involving harmonic numbers. Romeo Meštrović Copyright © 2013 Romeo Meštrović. All rights reserved. Wed, 19 Dec 2012 17:40:16 +0000 http://www.hindawi.com/isrn/combinatorics/2013/450627/ Following a suggestion of Cichoń and Macyna, binary search trees are generalized by keeping (classical) binary search trees and distributing incoming data at random to the individual trees. Costs for unsuccessful and successful search are analyzed, as well as the internal path length. Helmut Prodinger Copyright © 2013 Helmut Prodinger. All rights reserved. Wed, 19 Dec 2012 12:04:22 +0000 http://www.hindawi.com/isrn/combinatorics/2013/979487/ Let be the set of nonnegative integers. For a given set the representation functions , are defined as the number of solutions of the equation with condition , , respectively. In this paper, we prove that if and , then cannot hold for all sufficiently large integers where . Wen Yu and Min Tang Copyright © 2013 Wen Yu and Min Tang. All rights reserved.