Showing papers in "The Journal of Combinatorics in 2010"

Projective planes I

Abstract: Semiadditive rings are defined and their relationship with the projective planes is studied. Free semiadditive rings provide an analogue of the ring of integers and polynomials for the ternary rings. A structure theory for free semiadditive rings is developed. It is shown that each element of a large class of semiadditive rings is obtained from a quotient of a polynomial ring over integers by an additive subgroup, by twisting addition and multiplication. This class includes all planar ternary rings. These methods are being developed to study the well known conjectures that every finite projective plane with no proper subplane is isomorphic to a prime field plane and that the order of a finite projective plane is a power of a prime number. Applications to these problems will be discussed in part II.

The permanent of a square matrix

Abstract: We investigate the permanent of a square matrix over a field and calculate it using ways different from Ryser's formula or the standard definition. One formula is related to symmetric tensors and has the same efficiency O(2^mm) as Ryser's method. Another algebraic method in the prime characteristic case uses partial differentiation.

Characterization of graphs using domination polynomials

Abstract: Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)=@?"i"="1^nd(G,i)x^i, where d(G,i) is the number of dominating sets of G of size i. A root of D(G,x) is called a domination root of G. We denote the set of distinct domination roots by Z(D(G,x)). Two graphs G and H are said to be D-equivalent, written as G~H, if D(G,x)=D(H,x). The D-equivalence class of G is [G]={H:H~G}. A graph G is said to be D-unique if [G]={G}. In this paper, we show that if a graph G has two distinct domination roots, then Z(D(G,x))={-2,0}. Also, if G is a graph with no pendant vertex and has three distinct domination roots, then Z(D(G,x))@?{0,-2+/-2i,-3+/-3i2}. Also, we study the D-equivalence classes of some certain graphs. It is shown that if n=0,2(mod3), then C"n is D-unique, and if n=0(mod3), then [P"n] consists of exactly two graphs.

On distance-balanced graphs

Abstract: It is shown that the graphs for which the Szeged index equals @?G@?@?|G|^24 are precisely connected, bipartite, distance-balanced graphs. This enables us to disprove a conjecture proposed in [M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, S.G. Wagner, Some new results on distance-based graph invariants, European J. Combin. 30 (2009) 1149-1163]. Infinite families of counterexamples are based on the Handa graph, the Folkman graph, and the Cartesian product of graph. Infinite families of distance-balanced, non-regular graphs that are prime with respect to the Cartesian product are also constructed.

On a conjecture about the Szeged index

Abstract: Khalifeh, Yousefi-Azari, Ashrafi and Wagner [M.K. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, S.G. Wagner, Some new results on distance-based graph invariants, European J. Combin. 30 (2009) 1149-1163] conjectured that for a connected graph G on n vertices and m edges with Szeged index Sz, Sz=mn^2/4 if and only if G is a regular bipartite graph. In this note, we disprove this conjecture and then prove a stronger result from which it follows that the equality holds if and only if G is a transmission-regular bipartite graph.

Relations on Krasner (m,n)-hyperrings

Abstract: The aim of this research work is to define and characterize a new class of n-ary multialgebras that we call Krasner (m,n)-hyperrings. They are a generalization of canonical n-ary hypergroups, a generalization of (m,n)-rings, a generalization of hyperrings in the sense of Krasner and a subclass of the (m,n)-hyperrings. Also, the three isomorphism theorems of ring theory and Krasner hyperring theory are derived in the context of Krasner (m,n)-hyperrings.

Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs

Abstract: We establish relations between the bandwidth and the treewidth of bounded degree graphs G, and relate these parameters to the size of a separator of G as well as the size of an expanding subgraph of G. Our results imply that if one of these parameters is sublinear in the number of vertices of G then so are all the others. This implies for example that graphs of fixed genus have sublinear bandwidth or, more generally, a corresponding result for graphs with any fixed forbidden minor. As a consequence we establish a simple criterion for universality for such classes of graphs and show for example that for each @c>0 every n-vertex graph with minimum degree ([email protected])n contains a copy of every bounded-degree planar graph on n vertices if n is sufficiently large.

The q-tangent and q-secant numbers via continued fractions

Abstract: It is well known that the (-1)-evaluation of the enumerator polynomials of permutations (resp derangements) by the number of excedances gives rise to tangent numbers (resp secant numbers) Recently, two distinct q-analogues of the latter result have been discovered by Foata and Han, and Josuat-Verges, respectively In this paper, we will prove some general continued fraction expansion formulae, which permit us to give a unified treatment of Josuat-Verges' two formulae and also to derive a new q-analogue of the aforementioned formulae Our approach is based on a (p,q)-analogue of tangent and secant numbers via continued fractions and also the generating function of permutations with respect to the quintuple statistic consisting of fixed point number, weak excedance number, crossing number, nesting number and inversion number We also give a combinatorial proof of Josuat-Verges' formulae by using a new linear model of derangements

Two-orbit polyhedra from groups

Abstract: polytopes are combinatorial structures that generalize the classical notion of convex polytopes. In this paper particular attention is given to polytopes whose automorphism groups (or groups of symmetries) have exactly two orbits on the set of flags. Two-orbit polytopes of rank n are classified in terms of their local configuration of flags. Furthermore, a characterization of two-orbit polyhedra in terms of their automorphism group is given; that is, necessary and sufficient conditions for a group to be the automorphism group of a two-orbit polyhedron are found.

An extension of the bivariate chromatic polynomial

Abstract: K. Dohmen, A. Ponitz and P. Tittmann [K. Dohmen, A. Ponitz, P. Tittmann, A new two-variable generalization of the chromatic polynomial, Discrete Mathematics and Theoretical Computer Science 6 (2003), 69-90], introduced a bivariate generalization of the chromatic polynomial P(G,x,y) which subsumes also the independent set polynomial of I. Gutman and F. Harary [I. Gutman, F. Harary, Generalizations of the matching polynomial, Utilitas Mathematicae 24 (1983), 97-106] and the vertex-cover polynomial of F.M. Dong, M.D. Hendy, K.T. Teo and C.H.C. Little [F.M. Dong, M.D. Hendy, K.L. Teo, and C.H.C. Little, The vertex-cover polynomial of a graph, Discrete Mathematics 250 (2002), 71-78]. We first show that P(G,x,y) has a recursive definition with respect to three kinds of edge eliminations: edge deletion, edge contraction, and edge extraction, i.e. deletion of an edge together with its endpoints. Like in the case of deletion and contraction only [J.G. Oxley and D.J.A. Welsh, The Tutte polynomial and percolation, in: J.A. Bundy, U.S.R. Murty (Eds.), Graph Theory and Related Topics, Academic Press, London, 1979, pp. 329-339] it turns out that there is a most general, or as they call it, a universal polynomial satisfying such recurrence relations with respect to the three kinds of edge eliminations, which we call @x(G,x,y,z). We show that the new polynomial simultaneously generalizes, P(G,x,y), as well as the Tutte polynomial and the matching polynomial, We also give an explicit definition of @x(G,x,y,z) using a subset expansion formula. We also show that @x(G,x,y,z) can be viewed as a partition function, using counting of weighted graph homomorphisms. Furthermore, we expand this result to edge-labeled graphs as was done for the Tutte polynomial by T. Zaslavsky [T. Zaslavsky, Strong Tutte functions of matroids and graphs, Trans. Amer. Math. Soc. 334 (1992), 317-347] and by B. Bollobas and O. Riordan [B. Bollobas, O. Riordan, A Tutte polynomial for coloured graphs, Combinatorics, Probability and Computing 8 (1999), 45-94]. The edge-labeled polynomial @x"l"a"b(G,x,y,z,t@?) also generalizes the chain polynomial of R.C. Read and E.G. Whitehead Jr. [R.C. Read, E.G. Whitehead Jr., Chromatic polynomials of homeomorphism classes of graphs, Discrete Mathematics 204 (1999), 337-356]. Finally, we discuss the complexity of computing @x(G,x,y,z).

Planar graphs are 1-relaxed, 4-choosable

Abstract: We show that every planar graph G=(V,E) is 1-relaxed, 4-choosable. This means that, for every list assignment L that assigns a set of at least four colors to each vertex, there exists a coloring f such that f(v)@?L(v) for every vertex v@?V and each color class f^-^1(@a) of f induces a subgraph with maximum degree at most 1.

Legendre-Stirling permutations

Abstract: We first give a combinatorial interpretation of Everitt, Littlejohn, and Wellman's Legendre-Stirling numbers of the first kind. We then give a combinatorial interpretation of the coefficients of the polynomial (1-x)^3^k^+^1@?"n"="0^~{{n+kn}}x^n analogous to that of the Eulerian numbers, where {{nk}}are Everitt, Littlejohn, and Wellman's Legendre-Stirling numbers of the second kind. Finally we use a result of Bender to show that the limiting distribution of these coefficients as n approaches infinity is the normal distribution.

Combinatorial characterization of the Assur graphs from engineering

Abstract: We introduce the idea of Assur graphs, a concept originally developed and exclusively employed in the literature of the kinematics community. This paper translates the terminology, questions, methods and conjectures from the kinematics terminology for one degree of freedom linkages to the terminology of Assur graphs as graphs with special properties in rigidity theory. Exploiting the recent works in combinatorial rigidity theory we provide mathematical characterizations of these graphs derived from ‘minimal’ linkages. With these characterizations, we confirm a series of conjectures posed by Offer Shai, and offer techniques and algorithms to be exploited further in future work.

Products of linear forms and Tutte polynomials

Abstract: Let @D be a finite sequence of n vectors from a vector space over any field We consider the subspace of Sym(V) spanned by @?"v"@?"Sv, where S is a subsequence of @D A result of Orlik and Terao provides a doubly indexed direct sum decomposition of this space The main theorem is that the resulting Hilbert series is the Tutte polynomial evaluation T(@D;1+x,y) Results of Ardila and Postnikov, Orlik and Terao, Terao, and Wagner are obtained as corollaries

Hypergroups and n-ary relations

Abstract: In this paper we associate a hypergroupoid with an n-ary relation @r defined on a nonempty set H. We investigate when it is an H"v-group, a hypergroup or a join space. Then we determine some connections between this hypergroupoid and Rosenberg's hypergroupoid associated with a binary relation.

On the equivalence between real mutually unbiased bases and a certain class of association schemes

Abstract: Mutually unbiased bases (MUBs) in complex vector spaces play several important roles in quantum information theory. At present, even the most elementary questions concerning the maximum number of such bases in a given dimension and their construction remain open. In an attempt to understand the complex case better, some authors have also considered real MUBs, mutually unbiased bases in real vector spaces. The main results of this paper establish an equivalence between sets of real mutually unbiased bases and 4-class cometric association schemes which are both Q-bipartite and Q-antipodal. We then explore the consequences of this equivalence, constructing new cometric association schemes and describing a potential method for the construction of sets of real MUBs.

Minimal identifying codes in trees and planar graphs with large girth

Abstract: Let G be a finite undirected graph with vertex set V(G). If [email protected]?V(G), let N[v] denote the closed neighbourhood of v, i.e. v itself and all its adjacent vertices in G. An identifying code in G is a subset C of V(G) such that the sets N[v]@?C are nonempty and pairwise distinct for each vertex [email protected]?V(G). We consider the problem of finding the minimum size of an identifying code in a given graph, which is known to be NP-hard. We give a linear algorithm that solves it in the class of trees, but show that the problem remains NP-hard in the class of planar graphs with arbitrarily large girth.

A q-enumeration of alternating permutations

Abstract: A classical result of Euler states that the tangent numbers are an alternating sum of Eulerian numbers. A dual result of Roselle states that the secant numbers can be obtained by a signed enumeration of derangements. We show that both identities can be refined with the following statistics: the number of crossings in permutations and derangements, and the number of patterns 31-2 in alternating permutations. Using previous results of Corteel, Rubey, Prellberg, and the author, we derive closed formulas for both q-tangent and q-secant numbers. There are two different methods for obtaining these formulas: one with permutation tableaux and one with weighted Motzkin paths (Laguerre histories).

Rank-width and tree-width of H-minor-free graphs

Abstract: We prove that for any fixed r>=2, the tree-width of graphs not containing K"r as a topological minor (resp. as a subgraph) is bounded by a linear (resp. polynomial) function of their rank-width. We also present refinements of our bounds for other graph classes such as K"r-minor free graphs and graphs of bounded genus.

Association schemes based on attenuated spaces

Abstract: The subspaces of a given dimension in an attenuated space form the points of an association scheme. If the dimension is maximal, it is the bilinear forms graph, which has been thoroughly studied. In this paper, we discuss the general case, and obtain a family of symmetric association schemes which are a common generalization of the Grassmann schemes and the bilinear forms schemes. Moreover, all the parameters of the association scheme are computed.

A bijection between dominant Shi regions and core partitions

Abstract: It is well-known that Catalan numbers C"n=1n+1(2nn) count the number of dominant regions in the Shi arrangement of type A, and that they also count partitions which are both n-cores as well as (n+1)-cores. These concepts have natural extensions, which we call here the m-Catalan numbers and m-Shi arrangement. In this paper, we construct a bijection between dominant regions of the m-Shi arrangement and partitions which are both n-cores as well as (mn+1)-cores. The bijection is natural in the sense that it commutes with the action of the affine symmetric group.

Linking the Calkin-Wilf and Stern-Brocot trees

Abstract: Links between the Calkin-Wilf tree and the Stern-Brocot tree are discussed answering the questions: What is thejth vertex in thenth level of the Calkin-Wilf tree? and Where is the vertexrslocated in the Calkin-Wilf tree? A simple mechanism is described for converting the jth vertex in the nth level of the Calkin-Wilf tree into the jth entry in the nth level of the Stern-Brocot tree. We also provide a simple method for evaluating terms in the Hyperbinary sequence thus answering a challenge raised in Quantum in September 1997. We also examine successors and predecessors in both trees.

Parity considerations in Andrews-Gordon identities

Abstract: In 1974, Andrews discovered the generating function for the partitions of n considered in a theorem due to Gordon. In a more recent paper, he reconsidered this generating function and gave refinements where additional restrictions involving parities are imposed. A combinatorial construction for the partitions enumerated by the mentioned generating function is given. Some of the Andrews' refinements are proven combinatorially, and a conjecture of his is settled.

The generic rank of body-bar-and-hinge frameworks

Abstract: Tay [T.S. Tay, Rigidity of multi-graphs I Linking Bodies in n-space, J. Combin. Theory B 26 (1984) 95-112] characterized the multigraphs which can be realized as infinitesimally rigid d-dimensional body-and-bar frameworks. Subsequently, Tay [T.S. Tay, Linking (n-2)-dimensional panels in n-space II: (n-2,2)-frameworks and body and hinge structures, Graphs Combin. 5 (1989) 245-273] and Whiteley [W. Whiteley, The union of matroids and the rigidity of frameworks, SIAM J. Discrete Math. 1 (2) (1988) 237-255] independently characterized the multigraphs which can be realized as infinitesimally rigid d-dimensional body-and-hinge frameworks. We adapt Whiteley's proof technique to characterize the multigraphs which can be realized as infinitesimally rigid d-dimensional body-bar-and-hinge frameworks. More importantly, we obtain a sufficient condition for a multigraph to be realized as an infinitesimally rigid d-dimensional body-and-hinge framework in which all hinges lie in the same hyperplane. This result is related to a long-standing conjecture of Tay and Whiteley [T.S. Tay, W. Whiteley, Recent advances in the generic rigidity of structures, Structural Topology 9 (1984) 31-38] which would characterize when a multigraph can be realized as an infinitesimally rigid d-dimensional body-and-hinge framework in which all the hinges incident to each body lie in a common hyperplane. As a corollary we deduce that if a graph G has three spanning trees which use each edge of G at most twice, then its square can be realized as an infinitesimally rigid three-dimensional bar-and-joint framework.

Regular embeddings of Kn,n where n is a power of 2. II: The non-metacyclic case

Abstract: The aim of this paper is to complete a classification of regular orientable embeddings of complete bipartite graphs K"n","n, where n=2^e. The method involves groups G which factorise as a product G=XY of two cyclic groups of order n such that the two cyclic factors are transposed by an involutory automorphism. In particular, we give a classification of such groups G in the case where G is not metacyclic. We prove that for each n=2^e, e>=3, there are up to map isomorphism exactly four regular embeddings of K"n","n such that the automorphism group G preserving the surface orientation and the bi-partition of vertices is a non-metacyclic group, and that there is one such embedding when n=4.

A structural result for hypergraphs with many restricted edge colorings

Abstract: For k-uniform hypergraphs F and H and an integer r 2, let cr;F (H) denote the number of r-colorings of the set of hyperedges of H with no monochromatic copy of F and let cr;F (n) = maxH2Hn cr;F (H), where the maximum runs over the familyHn of all k-uniform hypergraphs on n vertices. Moreover, let ex(n;F ) be the usual Tur an function , i.e., the maximum number of hyperedges of an n-vertex k-uniform hypergraph which contains no copy of F . In this paper, we consider the question for determining cr;F (n) for arbitrary xed hypergraphs F and show

Contractibility and the Hadwiger Conjecture

Abstract: Consider the following relaxation of the Hadwiger Conjecture: For each t there exists N"t such that every graph with no K"t-minor admits a vertex partition into @[email protected][email protected]@? parts, such that each component of the subgraph induced by each part has at most N"t vertices. The Hadwiger Conjecture corresponds to the case @a=1, @b=-1 and N"t=1. Kawarabayashi and Mohar [K. Kawarabayashi, B. Mohar, A relaxed Hadwiger's conjecture for list colorings, J. Combin. Theory Ser. B 97 (4) (2007) 647-651. URL: http://dx.doi.org/10.1016/j.jctb.2006.11.002] proved this relaxation with @a=312 and @b=0 (and N"t a huge function of t). This paper proves this relaxation with @a=72 and @b=-32. The main ingredients in the proof are: (1) a list colouring argument due to Kawarabayashi and Mohar, (2) a recent result of Norine and Thomas that says that every sufficiently large (t+1)-connected graph contains a K"t-minor, and (3) a new sufficient condition for a graph to have a set of edges whose contraction increases the connectivity.

New identifying codes in the binary Hamming space

Abstract: Let F^n be the binary n-cube, or binary Hamming space of dimension n, endowed with the Hamming distance. For r>=1 and x@?F^n, we denote by B"r(x) the ball of radius r and centre x. A set C@?F^n is said to be an r-identifying code if the sets B"r(x)@?C, x@?F^n, are all nonempty and distinct. We give new constructive upper bounds for the minimum cardinalities of r-identifying codes in the Hamming space.

Davenport constant with weights

Abstract: Let [email protected]?{1,...,n-1} and let (x"1,...,x"t)@?Z^t be a sequence of integers with a maximal length such that for all (a"1,...,a"t)@?A^t,@?"1"@?"i"@?"ta"ix"[email protected]?0(modn). The authors show that for any sequence of integers (x"1,...,x"n"+"t)@?Z^n^+^t, there are b"1,...,b"[email protected]?A and [email protected]?k"1