Showing papers in "The Journal of Combinatorics in 2006"
TL;DR: The notions tree-depth and upper chromatic number of a graph are defined and it is shown that the upper chromatics number coincides with the maximal function which can be locally demanded in a bounded coloring of any proper minor closed class of graphs.
Abstract: We define the notions tree-depth and upper chromatic number of a graph and show their relevance to local-global problems for graph partitions. In particular we show that the upper chromatic number coincides with the maximal function which can be locally demanded in a bounded coloring of any proper minor closed class of graphs. The rich interplay of these notions is applied to a solution of bounds of proper minor closed classes satisfying local conditions. In particular, we prove the following result:For every graph M and a finite set F of connected graphs there exists a (universal) graph U = U(M, F) ∈ Forbh(F) such that any graph G ∈ Forbh(F) which does not have M as a minor satisfies G → U (i.e. is homomorphic to U).This solves the main open problem of restricted dualities for minor closed classes and as an application it yields the bounded chromatic number of exact odd powers of any graph in an arbitrary proper minor closed class. We also generalize the decomposition theorem of DeVos et al. [M. DeVos, G. Ding, B. Oporowski, D.P. Sanders, B. Reed, P. Seymour, D. Vertigan, Excluding any graph as a minor allows a low tree-width 2-coloring, J. Combin. Theory Ser. B 91 (2004) 25-41].
276 citations
TL;DR: A large self-dual class of simplicial complexes for which each member complex is contractible or homotopy equivalent to a sphere is introduced.
Abstract: We introduce a large self-dual class of simplicial complexes for which we show that each member complex is contractible or homotopy equivalent to a sphere. Examples of complexes in this class include independence and dominance complexes of forests, pointed simplicial complexes, and their combinatorial Alexander duals.
85 citations
TL;DR: This paper proposes a new method for determining whether a family of graphs (which have no special properties) are DS with respect to their generalized spectra, obtained by employing some arithmetic properties of a certain matrix associated with a graph.
Abstract: A graph G is said to be determined by its spectrum (DS for short), if any graph having the same spectrum as G is necessarily isomorphic to G. One important topic in the theory of graph spectra is how to determine whether a graph is DS or not. The previous techniques used to prove a graph to be DS heavily rely on some special properties of the spectrum of the given graph. They cannot be applied to general graphs. In this paper, we propose a new method for determining whether a family of graphs (which have no special properties) are DS with respect to their generalized spectra. The method is obtained by employing some arithmetic properties of a certain matrix associated with a graph. Numerical examples are further given to illustrate the effectiveness of the proposed method.
67 citations
TL;DR: Most of the recent and often surprising results on packings of congruent spheres in d-dimensional spaces of constant curvature are surveyed, along with some of the "most wanted" research problems.
Abstract: In this paper we survey most of the recent and often surprising results on packings of congruent spheres in d-dimensional spaces of constant curvature. The topics discussed are as follows: - Hadwiger numbers of convex bodies and kissing numbers of spheres; - touching numbers of convex bodies; - Newton numbers of convex bodies; - one-sided Hadwiger and kissing numbers; - contact graphs of finite packings and the combinatorial Kepler problem; - isoperimetric problems for Voronoi cells, the strong dodecahedral conjecture and the truncated octahedral conjecture; - the strong Kepler conjecture; - bounds on the density of sphere packings in higher dimensions; - solidity and uniform stability. Each topic is discussed in details along with some of the "most wanted" research problems.
62 citations
TL;DR: It is shown that for all r ≥ 1, any r-identifying code of the cycle Cn has cardinality at least gcd(2r + 1, n) ⌈n/2gcd( 2r+ 1,n)⌉, and this lower bound is enough to solve the case n even.
Abstract: In this paper we deal with identifying codes in cycles. We show that for all r ≥ 1, any r-identifying code of the cycle Cn has cardinality at least gcd(2r + 1, n) ⌈n/2gcd(2r+ 1,n)⌉. This lower bound is enough to solve the case n even (which was already solved in [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European Journal of Combinatorics 25 (7) (2004) 969-987]), but the case n odd seems to be more complicated. An upper bound is given for the case n odd, and some special cases are solved. Furthermore, we give some conditions on n and r to attain the lower bound.
58 citations
TL;DR: In this paper, structural aspects of lattice path matroids are studied in terms of fundamental flats, which are special connected flats from which one can recover the paths that define the matroid.
Abstract: This paper studies structural aspects of lattice path matroids. Among the basic topics treated are direct sums, duals, minors, circuits, and connected flats. One of the main results is a characterization of lattice path matroids in terms of fundamental flats, which are special connected flats from which one can recover the paths that define the matroid. We examine some aspects related to key topics in the literature of transversal matroids and we determine the connectivity of lattice path matroids. We also introduce notch matroids, a minor-closed, dual-closed subclass of lattice path matroids, and we find their excluded minors.
54 citations
TL;DR: In this paper, all Cohen-Macaulay polymatroidal ideals are classified into three classes: the principal ideals, the Veronese ideals, and the square-free Veroneses ideals.
Abstract: All Cohen-Macaulay polymatroidal ideals are classified. The Cohen-Macaulay polymatroidal ideals are precisely the principal ideals, the Veronese ideals, and the square-free Veronese ideals.
53 citations
TL;DR: The equality can be modified to an inequality in such a way that it characterizes isometric subgraphs of Hamming graphs and simplifies recognition of these graphs and computation of their average distance.
Abstract: Average distance of a graph is expressed in terms of its canonical metric representation. The equality can be modified to an inequality in such a way that it characterizes isometric subgraphs of Hamming graphs. This approach simplifies recognition of these graphs and computation of their average distance.
51 citations
TL;DR: In this article, the exact rank of connection matrices was determined and it was shown that if two k-tuples of nodes behave in the same way from the point of view of graph homomorphisms, then they are equivalent under the automorphism group.
Abstract: Connection matrices were introduced in [M. Freedman, L. Lovasz, A. Schrijver, Reflection positivity, rank connectivity, and homomorphism of graphs (MSR Tech Report # MSR-TR-2004-41) ftp://ftp.research.microsoft.com/pub/tr/TR-2004-41.pdf], where they were used to characterize graph homomorphism functions. The goal of this note is to determine the exact rank of these matrices. The result can be rephrased in terms of the dimension of graph algebras, also introduced in the same paper. Yet another version proves that if two k-tuples of nodes behave in the same way from the point of view of graph homomorphisms, then they are equivalent under the automorphism group.
48 citations
TL;DR: A classification of semifield planes of order q4 with kernel Fq2 and center Fq is given, proving the conjecture that for q an odd prime, this proves the conjecture stated in Cordero-Figueroa's conjecture.
Abstract: A classification of semifield planes of order q4 with kernel Fq2 and center Fq is given. For q an odd prime, this proves the conjecture stated in [M. Cordero, R. Figueroa, On the semifield planes of order 54 and dimension 2 over the kernel, Note Mat. (in press)]. Also, we extend the classification of semifield planes lifted from Desarguesian planes of order q2, q odd, obtained in [M. Cordero, R. Figueroa, On some new classes of semifield planes, Osaka J. Math. 30 (1993) 171-178], to the even characteristic case.
48 citations
TL;DR: In this paper, the authors provide an algebraic setting for cumulants and factorial moments via the classical umbral calculus, which is related to logarithmic power series.
Abstract: We provide an algebraic setting for cumulants and factorial moments via the classical umbral calculus. Our main tools are the compositional inverse of the unity umbra, this being related to logarithmic power series, and a new umbra here introduced, the singleton umbra. We develop formulae that express cumulants, factorial moments and central moments as umbral functions.
TL;DR: In this paper, a new family of strongly regular graphs, called the general symplectic graphs Sp(2v, q), associated with nonsingular alternate matrices is introduced, and the parameters of these graphs, their chromatic numbers as well as their groups of graph automorphisms are determined.
Abstract: A new family of strongly regular graphs, called the general symplectic graphs Sp(2v, q), associated with nonsingular alternate matrices is introduced. Their parameters as strongly regular graphs, their chromatic numbers as well as their groups of graph automorphisms are determined.
TL;DR: It is shown that for any integer α ≥ 8, there exists a connected well-covered graph G with α = α(G), whose independence polynomial is not unimodal.
Abstract: A graph G is well-covered if all its maximal stable sets have the same size, denoted by α(G) [M.D. Plummer, Some covering concepts in graphs, Journal of Combinatorial Theory 8 (1970) 91-98]. If sk denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then I(G; x) = Σk=0α(G) skxk is the independence polynomial of G [I. Gutman, F. Harary, Generalizations of the matching polynomial, Utilitas Mathematica 24 (1983) 97-106]. J.I. Brown, K. Dilcher and R.J. Nowakowski [Roots of independence polynomials of well-covered graphs, Journal of Algebraic Combinatorics 11 (2000) 197-210] conjectured that I(G; x) is unimodal (i.e., there is some j ∈ {0, 1,..., α(G)} such that s0 ≤ ... ≤ sj-1 ≤ sj ≥ sj+1 ≥ ... ≥ sα(G)) for any well-covered graph G. T.S. Michael and W.N. Traves [Independence sequences of well-covered graphs: non-unimodality and the roller-coaster conjecture, Graphs and Combinatorics 19 (2003) 403-411] proved that this assertion is true for α(G) ≤ 3, while for α(G) ∈ {4,5,6,7} they provided counterexamples.In this paper we show that for any integer α ≥ 8, there exists a connected well-covered graph G with α = α(G), whose independence polynomial is not unimodal. In addition, we present a number of sufficient conditions for a graph G with α(G) ≤ 6 to have the unimodal independence polynomial.
TL;DR: For m > n ≥ 0 and 1 ≤ d ≤ m, it is shown that the q-Euler number E2m(q) is congruent to qm-n E2n (q) mod (1 + qd) if and only if m ≡ n mod d.
Abstract: For m > n ≥ 0 and 1 ≤ d ≤ m, it is shown that the q-Euler number E2m(q) is congruent to qm-n E2n(q) mod (1 + qd) if and only if m ≡ n mod d. The q-Salie number S2n(q) is shown to be divisible by (1 + q2r+1)⌊n/2r+1⌋ for any r ≥ 0. Furthermore, similar congruences for the generalized q-Euler numbers are also obtained, and some conjectures are formulated.
TL;DR: A theory of isometric subgraphs of hypercubes for which a certain inheritance of isometry plays a crucial role is developed, demonstrating that the concept of lopsidedness in its various diguises si most natural and cersatile in combinatorics.
Abstract: We develop a theory of isometric subgraphs of hypercubes for which a certain inheritance of isometry plays a crucial role. It is well known that median graphs and closely related graphs embedded in hypercubes bear geometric features that involve realizations by solid cubical complexes or are expressed by Euler-type counting formulae for cubical faces. Such properties can also be established for antimatroids, and in fact, a straightforward generalization ("conditional antimatroid") captures this concept as well as median convexity. The key ingredient for the cube counting formulae that work in conditional antimatroids is a simple cube projection property, which, when letting sets be encoded by sign vectors, is seen to be invariant under sign switches and guarantees linear independence of the corresponding sign vectors. It then turns out that a surprisingly elementary calculus of projection and lifting gives rise to a plethora of equivalent characterizations of set systems bearing these properties, which are not necessarily closed under intersections (and thus are more general than conditional antimatroids). One of these descriptions identifies these particular set systems alias sets of sign vectors as the lopsided sets originally introduced by Lawrence in order to investigate the subgraphs of the n-cube that encode the intersection pattern of a given convex set K with the closed orthans of the n dimensional Euclidean space. This demonstrates that the concept of lopsidedness in its various diguises si most natural and cersatile in combinatorics.
TL;DR: It is shown that the smallest possible density of a locating-dominating set in the king grid equals 1/5 and in the hexagonal mesh 1/3.
Abstract: Determining a malfunctioning component in a processor network gives the motivation for locating-dominating sets. It is shown that the smallest possible density of a locating-dominating set in the king grid equals 1/5 and in the hexagonal mesh 1/3. Moreover, we discuss a natural modification of locating-dominating sets.
TL;DR: A general method for finding elementary abelian regular covering projections of finite connected graphs is applied to the Petersen graph and a complete list of pairwise nonisomorphic elementary abELian covers admitting a lift of a vertex-transitive group of automorphisms is given.
Abstract: A general method for finding elementary abelian regular covering projections of finite connected graphs is applied to the Petersen graph. As a result, a complete list of pairwise nonisomorphic elementary abelian covers admitting a lift of a vertex-transitive group of automorphisms is given. The resulting graphs are explicitly described in terms of voltage assignments.
TL;DR: It is shown that any one-regular and 4-valent Cayley graph X = Cay(G, S) of dihedral groups G is normal except that n = 4s, and X ≃ Cay (G, {a, a-1, aib, a -ib}), where i2 ≡ ± 1 (mod 2s), 2 ≤ i ≤ 2s - 2.
Abstract: A Cayley graph X = Cay(G, S) of group G is said to be normal if R(G) is normal in Aut(X). Let G = «a, b | an = b2 = 1 », S be a generating set of G, |S| = 4. In this paper we show that any one-regular and 4-valent Cayley graph X = Cay(G, S) of dihedral groups G is normal except that n = 4s, and X ≃ Cay(G, {a, a-1, aib, a-ib}), where i2 ≡ ± 1 (mod 2s), 2 ≤ i ≤ 2s - 2.
TL;DR: The proofs are based on Szemeredi's Regularity Lemma together with the Simonovits Stability Theorem, and provide one of the growing number of examples of a precise result proved by applying the Regularities Lemma.
Abstract: Fix a 2-coloring Hk + 1 of the edges of a complete graph Kk + 1. Let C(n, Hk + 1) denote the maximum possible number of distinct edge-colorings of a simple graph on n vertices with two colors, which contain no copy of Kk + 1 colored exactly as Hk + 1. It is shown that for every fixed k and all n > n0(k), if in the colored graph Hk + 1 both colors were used, then C(n, Hk + 1) = 2tk(n), where tk(n) is the maximum possible number of edges of a graph on n vertices containing no K k + 1. The proofs are based on Szemeredi's Regularity Lemma together with the Simonovits Stability Theorem, and provide one of the growing number of examples of a precise result proved by applying the Regularity Lemma.
TL;DR: Answering a question of Terence Tao, the following improvement of (1) is shown: the classical uncertainty inequality asserts that if f ≠ 0 then |supp(f)| ċ |supp (f)| ≥ |G|.
Abstract: Let G be a finite abelian group of order n. For a complex valued function f on G let fdenote the Fourier transform of f. The classical uncertainty inequality asserts that if f ≠ 0 then |supp(f)| ċ |supp(f)| ≥ |G|. (1) Answering a question of Terence Tao, the following improvement of (1) is shown: Theorem. Let d1 < d2 be two consecutive divisors of n. If d1 ≤ k = |supp(f)| ≤ d2 then |supp(f)| ≥ n/d1d2(d1 + d2 - k).
TL;DR: The concept of Hadamard ideal is introduced, to systematize the application of computational algebra methods to the construction of HadAmard matrices with two circulant cores, given by Fletcher, Gysin and Seberry.
Abstract: We apply computational algebra methods to the construction of Hadamard matrices with two circulant cores, given by Fletcher, Gysin and Seberry. We introduce the concept of Hadamard ideal, to systematize the application of computational algebra methods for this construction. We use the Hadamard ideal formalism to perform exhaustive search constructions of Hadamard matrices with two circulant cores for the twelve orders 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52. The total number of such Hadamard matrices is proportional to the square of the parameter. We use the Hadamard ideal formalism to compute the proportionality constants for the twelve orders listed above. Finally, we use the Hadamard ideal formalism to improve the lower bounds for the number of inequivalent Hadamard matrices for the seven orders 44, 48, 52, 56, 60, 64, 68.
TL;DR: It is shown that admitting an interpolating labelling is again equivalent to possessing a maximal chain of left modular elements in the setting of an arbitrary bounded poset.
Abstract: It is known that a graded lattice of rank n is supersolvable if and only if it has an EL-labelling where the labels along any maximal chain are exactly the numbers 1, 2, .....n without repetition. These labellings are called Sn EL-labellings, and having such a labelling is also equivalent to possessing a maximal chain of left modular elements. In the case of an ungraded lattice, there is a natural extension of Sn EL-labellings, called interpolating labellings. We show that admitting an interpolating labelling is again equivalent to possessing a maximal chain of left modular elements. Furthermore, we work in the setting of an arbitrary bounded poset as all the above results generalize to this case.
TL;DR: It is proved that there is a semiregular automorphism of order tending to infinity with n and there is one of order greater than 2.
Abstract: An old conjecture of Marusic, Jordan and Klin asserts that any finite vertex-transitive graph has a non-trivial semiregular automorphism. Marusic and Scapellato proved this for cubic graphs. For these graphs, we make a stronger conjecture, to the effect that there is a semiregular automorphism of order tending to infinity with n. We prove that there is one of order greater than 2.
TL;DR: This paper gives a generic group theoretic construction of homogeneous factorisations of arbitrary graphs and digraphs and shows that all homogeneity factorisations can be constructed in this way.
Abstract: A homogeneous factorisation (M, G, Γ, P) is a partition P of the arc set of a digraph Γ such that there exist vertex-transitive groups M < G ≤ Aut(Γ) such that M fixes each part of P setwise while G acts transitively on P. Homogeneous factorisations of complete graphs have previously been studied by the second and fourth authors, and are a generalisation of vertex-transitive self-complementary digraphs. In this paper we initiate the study of homogeneous factorisations of arbitrary graphs and digraphs. We give a generic group theoretic construction and show that all homogeneous factorisations can be constructed in this way. We also show that the important homogeneous factorisations to study are those where G acts transitively on the set of arcs of Γ, M is a normal subgroup of G and G/M is a cyclic group of prime order.
TL;DR: The Terwilliger algebra of a Hamming scheme H(d, q), which can be described as symmetric d-tensors on the τ-algebra of H(1, q) which are all isomorphic for q > 2 is computed.
Abstract: We compute the Terwilliger algebra of a Hamming scheme H(d, q). This algebra can be described as symmetric d-tensors on the τ-algebra of H(1, q) which are all isomorphic for q > 2. We give the decomposition into simple bilateral ideals using the representation theory of GL(2, C).
TL;DR: In this study, all circuits in the suborbital graph for the normalizer of Γ0(m) when m is a square-free positive integer are characterized and a conjecture concerning theSuborbital graphs is proposed.
Abstract: In this study, we characterize all circuits in the suborbital graph for the normalizer of Γ0(m) when m is a square-free positive integer. We propose a conjecture concerning the suborbital graphs.
TL;DR: It is proved that the construction of a maximum vertex independence set in a benzenoid is similar with the dual paths between pentagonal faces replaced by dual circuits through the outside face.
Abstract: We explore the structure of the maximum vertex independence sets in fullerenes: plane trivalent graphs with pentagonal and hexagonal faces. At the same time, we will consider benzenoids: plane graphs with hexagonal faces and one large outer face. In the case of fullerenes, a maximum vertex independence set may constructed as follows: (i) Pair up the pentagonal faces. (ii) Delete the edges of a shortest path in the dual joining the paired faces to get a bipartite subgraph of the fullerene. (iii) Each of the deleted edges will join two vertices in the same cell of the bipartition; eliminating one endpoint of each of the deleted edges results in two independent subsets.The main part of this paper is devoted to showing that for a properly chosen pairing, the larger of these two independent subsets will be a maximum independent set. We also prove that the construction of a maximum vertex independence set in a benzenoid is similar with the dual paths between pentagonal faces replaced by dual circuits through the outside face. At the end of the paper, we illustrate this method by computing the independence number for each of the icosahedral fullerenes.
TL;DR: In this paper, it was shown that Frankl's conjecture holds for families containing three 3-subsets of a 5-set, four 3-set of a 6-set and eight 4-subset of an 8-set.
Abstract: A family of sets A is said to be union-closed if {A∪B : A, B ∈ A} ⊂ A. Frankl's conjecture states that given any finite union-closed family of sets, not all empty, there exists an element contained in at least half of the sets. Here we prove that the conjecture holds for families containing three 3-subsets of a 5-set, four 3-subsets of a 6-set, or eight 4-subsets of a 6-set, extending work of Poonen and Vaughan. As an application we prove the conjecture in the case that the largest set has at most nine elements, extending a result of Gao and Yu. We also pose several open questions.
TL;DR: It is shown that these q-GenocchiNumbers have interesting combinatorial interpretations in the classical models for Genocchi numbers such as alternating pistols, alternating permutations, non-intersecting lattice paths and skew Young tableaux.
Abstract: A new q-analog of Genocchi numbers is introduced through a q-analog of Seidel's triangle associated with Genocchi numbers. It is then shown that these q-Genocchi numbers have interesting combinatorial interpretations in the classical models for Genocchi numbers such as alternating pistols, alternating permutations, non-intersecting lattice paths and skew Young tableaux.
TL;DR: Kashiwara-Nakashima combinatorics of crystal graphs associated with the roots systems Bn and Dn are used to extend the results of Lecouvey and Morris and derive explicit formulas for Kλµ(q) when λ≤ 3,orn = 2 and µ=0 from the cyclage graph structure on tableaux of type Cn.
Abstract: We use Kashiwara-Nakashima combinatorics of crystal graphs associated with the roots systems Bn and Dn to extend the results of Lecouvey [C. Lecouvey, Kostka-Foulkes polynomials, cyclage graphs and charge statistics for the root system Cn, J. Algebraic Combin. (in press)] and Morris [A.-O. Morris, The characters of the group GL(n, q), Math. Z. 81 (1963) 112-123] by showing that Morris-type recurrence formulas also exist for the orthogonal root systems. We derive from these formulas a statistic on Kashiwara-Nakashima tableaux of types Bn, Cn and Dn generalizing the Lascoux-Schutzenberger charge and from which it is possible to compute the Kostka-Foulkes polynomials Kλµ(q) under certain conditions on (λµ), This statistic is different from that obtained in Lecouvey [C. Lecouvey, Kostka-Foulkes polynomials, cyclage graphs and charge statistics for the root system Cn,, J. Algebraic Combin. (in press)] from the cyclage graph structure on tableaux of type Cn. We show that such a structure also exists for the tableaux of types Bn and Dn but cannot be related in a simple way to the Kostka-Foulkes polynomials. Finally we give explicit formulas for Kλµ(q) when λ≤ 3,orn = 2 and µ=0.