pinecone/core-2020-05-10-deduplication huggingface dataset:
Oriented Rank Three Matroids and Projective Planes

AbstractRecently, Goodman et al. have proven two conjectures by Grünbaum right, showing that any arrangement of pseudolines in the plane can be embedded into a flat projective plane and that there exists a universal topological projective plane in which every arrangement of pseudolines is stretchable. By Folkman and Lawrence’s theorem , this plane contains every finite (simple) oriented rank three matroid. In this paper, we will also consider embeddings of oriented rank three matroids into topological projective planes, but we will take a quite different viewpoint: we shall show that there exists a projective plane Π that contains the combinatorial geometry of every finite, orientable rank three matroidMn , such that any choice of orientations χnof theMn , n∈N, extends to an orientation χ ofΠ . Furthermore, these orientations correspond to archimedean orderings of Π, hence the reorientation classes of every finite rank three matroid can be studied by the set of archimedian orderings of Π. Since, by a celebrated result of Prieß-Crampe , any archimedian projective plane can be completed and thus embedded into a flat projective plane, our results yield another proof of Grünbaum’s conjectures and a new proof of the rank three case of Folkman and Lawrence’s theorem
Subsets with a Small Sum II: the Critical Pair Problem

AbstractA basic problem raised by Kneser was to describe all the finite subsets A, B of an abelian group G such that | A+B | ≤ | A | + | B | − 1. Let G be an abelian group. The description of the pairsA , B⊂G such that | A+B | = | A | + | B | − 1 < |G | was considered in additive group theory. Vosper (1956) solved this problem completely for groups with a prime order. Kempermann’s theory for small sums describes the structure of these pairs, ifA+B is aperiodic or if there exists a uniquely expressible element inA+B. In this paper we study the same question with a fixed subset B satisfying the inequality: for all A such that 1 ≤ | A | <∞, |A+B | ≥min(| G |, | A | + |B | − 1). We obtain a recursive description for the subsets A such that |A+B | ≤ | A | + | B | − 1. As corollary of our description, we obtain the following result which implies some limitations of Kempermann’s theory. Suppose that B is neither a coprogression nor almost periodic and that 2 ≤ |A | ≤ | G | − | B | − 1. If |A+B | = | A | + | B | − 1, then A is periodic and A+B contains no unique expression elements. The results obtained in this section are strongly based on those obtained in Part I
Subgraphs with Restricted Degrees of their Vertices in Large Polyhedral Maps on Compact Two-manifolds

AbstractLet k≥ 2, be an integer and M be a closed two-manifold with Euler characteristic χ(M) ≤ 0. We prove that each polyhedral map G onM , which has at least (8 k2+ 6 k− 6)|χ (M)| vertices, contains a connected subgraph H of order k such that every vertex of this subgraph has, in G, the degree at most 4 k+ 4. Moreover, we show that the bound 4k+ 4 is best possible. Fabrici and Jendrol’ proved that for the sphere this bound is 10 ifk= 2 and 4 k+ 3 if k≥ 3. We also show that the same holds for the projective plane
Generalized Quadrangles with a Spread of Symmetry

AbstractWe present a common construction for some known infinite classes of generalized quadrangles. Whether this construction yields other (unknown) generalized quadrangles is an open problem. The class of generalized quadrangles obtained this way is characterized in two different ways. On the one hand, they are exactly the generalized quadrangles having a spread of symmetry. On the other hand, they can be characterized in terms of the group of projectivities with respect to a spread. We explore some properties of these generalized quadrangles. All these results can be applied to the theory of the glued near hexagons, a class of near hexagons introduced by the author in De Bruyn (1998) On near hexagons and spreads of generalized quadrangles, preprint
Graded Characters of Modules Supported in the Closure of a Nilpotent Conjugacy Class

AbstractThis is a combinatorial study of the Poincaré polynomials of isotypic components of a natural family of graded GL(n)-modules supported in the closure of a nilpotent conjugacy class. These polynomials generalize the Kostka–Foulkes polynomials and are q -analogues of Littlewood–Richardson coefficients. The coefficients of two-column Macdonald–Kostka polynomials also occur as a special case. It is conjectured that these q -analogues are the generating function of so-called catabolizable tableaux with the charge statistic of Lascoux and Schützenberger. A general approach for a proof is given, and is completed in certain special cases including the Kostka–Foulkes case. Catabolizable tableaux are used to prove a characterization of Lascoux and Schützenberger for the image of the tableaux of a given content under the standardization map that preserves the cyclage poset
On Hering Decomposition ofDKnInduced by Group Actions on Conjugacy Classes

AbstractA Hering decomposition of type t and order n is a factorization of the complete directed graph DKninto arc disjoint t -circuits for which the following conditions hold: (1) The vertices and arcs of DKnare partitioned into arc disjoint parallel classes, each containing n−1 t circuits and a single vertex. (2) Any two parallel classes have exactly one edge (undirected arc) in common. For each parallel class Pi, 1 ≤i≤n there corresponds a permutationσi acting on the letter set {1, 2,⋯ , n } as follows:σi (k) =l iff (k,l ) is an arc of Pi. Let G be the group generated byσ1 , σ2,⋯ , σn. The given configuration is inner transitive if G permutes the parallel classes transitively. In this paper we study the interaction between an inner-transitive Hering configuration and the structure of the corresponding group G. In particular, we classify all the pairs (n, t) for which an inner-transitive Hering configuration of type t and order n exists
Non-splitting Abelian (4 t, 2, 4 t, 2 t) Relative Difference Sets and Hadamard Cocycles

AbstractUsing cohomology we show that in studying the existence of an abelian non-splitting (4 t, 2, 4 t, 2 t) relative difference set, D, we can assume the groups in question have a certain simple form. We obtain an explicit constructive equivalence between generalized perfect binary arrays and cocycles that define Hadamard matrices and thereby show directly that the existence of D corresponds to that of a symmetric Hadamard matrix of a certain form. This extends the well-known equivalence in the case of splitting relative difference sets
Counting Pattern-free Set Partitions I: A Generalization of Stirling Numbers of the Second Kind

AbstractA partition u of [ k ] = {1, 2,⋯ , k } is contained in another partition v of [ l ] if [ l ] has a k -subset on which v inducesu . We are interested in counting partitions v not containing a given partition u or a given set of partitions R. This concept is related to that of forbidden permutations. A strengthening of the Stanley–Wilf conjecture is proposed. We prove that the generating function (GF) counting v is rational if (i)R is finite and the number of parts of v is fixed or if (ii)u has only singleton parts and at most one doubleton part. In fact, (ii) is an application of (i). As another application of (i) we prove that for each k the GF counting partitions with k pairs of crossing parts belongs toZ(1 − 4 x)
Adjacency, Inseparability, and Base Orderability in Matroids

AbstractTwo elements in an oriented matroid are inseparable if they have either the same sign in every signed circuit containing them both or opposite signs in every signed circuit containing them both. Two elements of a matroid are adjacent if there is no M(K4)-minor using them both, and in which they correspond to a matching ofK4 . We prove that two elements e, f of an oriented matroid are inseparable if and only ife , f are inseparable in every M(K4) orU42 -minor containing them. This provides a link between inseparability in oriented matroids (introduced by Bland and Las Vergnas) and adjacency in binary matroids (introduced by Seymour). We define the concepts of base orderable and strongly base orderable subsets of a matroid, generalizing the definitions of base orderable and strongly base orderable matroids. Strongly base orderable subsets can be used to obtain packing and covering results, generalizing results of Davies and McDiarmid, as was shown in a previous paper. In this paper, we prove that any pairwise inseparable subset of an oriented matroid is base orderable. For binary matroids we derive the following characterization: a subset is strongly base orderable if and only if it is pairwise adjacent
Bulky Subgraphs of the Hypercube

AbstractLet Qdbe the d -dimensional hypercube on 2dvertices, and let G be its induced subgraph. We say that G is simple-majority if |G | > 2d−1and G is bulky if it is connected and meets every facet of Qd. We show that every simple-majority G has a bulky subgraph. This was conjectured by Alon, Seymour and Thomas (A separator theorem for non-planar graphs, J. Am. Math. Soc. 3 (1990) 801–808). Further, we show that such a subgraph can be chosen on d+ 1 vertices ifd≤ 5 and on fewer than 6 × 1.5d−5vertices if d≥ 6
Piercing Balls Sitting on a Table by a Vertical Line

AbstractLet Fnbe a family of disjoint n balls all sitting on a fixed horizontal table T. Let ℓ denote a vertical line that meets T. We prove that if ℓ meets 2 k+ 1 balls in Fn, then the radius of the smallest ball among the 2k+ 1 balls is at most (2 −3)ktimes the radius of the biggest ball among the 2 k+ 1 balls. Using this result we prove that for anyFn the average number of balls an ℓ meets is at most logn+o(1). A similar result for a two-dimensional version is also given together with a lower bound of the least upper bound
Completing some Partial Latin Squares

AbstractWe show that any partial 3 r× 3 r Latin square whose filled cells lie in two disjoint r×r sub-squares can be completed. We do this by proving the more general result that any partial 3 r by 3 r Latin square, with filled cells in the top left 2r× 2 r square, for which there is a pairing of the columns so that in each row there is a filled cell in at most one of each matched pair of columns, can be completed if and only if there is some way to fill the cells of the top left 2 r× 2 r square
A Lower Bound on Blocking Semiovals

AbstractA semioval in a projective plane Π is a set S of points such that for every pointP∈S, there exists a unique line ℓ of Π such thatℓ∩S= { P }. In other words, at every point of S, there exists a unique tangent line. A blocking set in Π is a set B of points such that every line ofΠ contains at least one point of B, but is not entirely contained in B. Combining these notions, we obtain the concept of a blocking semioval, a set of points in a projective plane which is both a semioval and a blocking set. Batten indicated applications of such sets to cryptography, which motivates their study. In this paper, we give some lower bounds on the size of a blocking semioval, and discuss the sharpness of these bounds
On Saturating Sets in Small Projective Geometries

AbstractA set of points, S⊆PG(r, q), is said to be ϱ -saturating if, for any point x∈PG(r, q), there exist ϱ+ 1 points in S that generate a subspace in which x lies. The cardinality of a smallest possible set S with this property is denoted by k(r, q,ϱ ). We give a short survey of what is known about k(r, q, 1) and present new results for k(r, q, 2) for small values of r and q. One construction presented proves that k(5, q, 2) ≤ 3 q+ 1 forq= 2, q≥ 4. We further give an upper bound onk (ϱ+ 1, pm, ϱ)
Decomposition andl1-Embedding of Weakly Median Graphs

AbstractWeakly median graphs, being defined by interval conditions and forbidden induced subgraphs, generalize quasi-median graphs as well as pseudo-median graphs. It is shown that finite weakly median graphs can be decomposed with respect to gated amalgamation and Cartesian multiplication into 5-wheels, induced subgraphs of hyperoctahedra (alias cocktail party graphs), and 2-connected bridged graphs not containing K4or K1,1,3as an induced subgraph. As a consequence one obtains that every finite weakly median graph is l1-embeddable, that is, it embeds as a metric subspace into some Rnequipped with the 1-norm
A Characterization of Minimizable Metrics in the Multifacility Location Problem

AbstractIn the minimum 0-extension problem (a version of the multifacility location problem), one is given a metric m on a subset X of a finite set V and a non-negative function c on the unordered pairs of elements of V. The objective is to find a semimetric m′ on V that minimizes the inner product c·m′, provided that m′ coincides with m within X and each element of V is at zero distance from X. For m fixed, this problem is solvable in strongly polynomial time if m is minimizable, which means that for any superset V and function c, the minimum objective value is equal to that in the corresponding linear relaxation. In , Karzanov showed that the path metric of a graph H is minimizable if and only if all isometric cycles of H have length four and the edges of H can be oriented so that non-adjacent edges in each 4-cycle have opposite orientations along the cycle (such graphs are called frames in ). Extending this result to general metrics m, we show that m is minimizable if and only if m is modular and its underlying graph is a frame
Set Systems and Dissimilarities

AbstractDifferent one–one correspondences exist between classes of indexed clustering structures (such as indexed hierarchies) and classes of dissimilarities (such as ultrametrics). Following the line developed in previous works (i.e., Johnson (1967), Diday (1984, 1986), Bertrand and Diday (1991), Batbedat (1988), Bandelt and Dress (1994)), we associate each pseudo-dissimilarityδ defined on a finite set E with the set system of all maximal linked subsets of E. This provides a one–one correspondence that maps the set of pseudo-dissimilarities onto the collection of all the valued set systems (S, f) that satisfy two conditions. One of these conditions was introduced by Batbedat (1988) and is related to the characterization of 2-conformity in hypergraphs (see also Bandelt and Dress (1994)). We introduce the other condition which requires the index f to be a weak index. Our approach includes a characterization of the valued set systems (S, f) such that S contains (respectively is contained in) the set of all finite nonempty intersections of maximal linked subsets with respect to the pseudo-dissimilarity induced by the pair (S, f)
Uniform Partitions of 3-space, their Relatives and Embedding

AbstractWe review 28 uniform partitions of 3-space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic latticeZn. We also consider some relatives of those 28 partitions, including Archimedean 4-polytopes of Conway–Guy, non-compact uniform partitions, Kelvin partitions and those with a unique vertex figure (i.e., Delaunay star). Among the latter ones we indicate two continuums of aperiodic tilings by semi-regular 3-prisms with cubes or with regular tetrahedra and regular octahedra. In the process many new partitions are added to the incomplete cases considered here
On the Geometry of Baer Subplanes inPG(2,4)

AbstractLet C be an L3(4)-orbit of Baer subplanes in PG(2, 4). We define a graph Γ on C where two subplanes are called adjacent if their intersection contains only one point. We determine Γ and the action of L3(4) is analysed in detail. It is shown that Γ is the collinearity graph of the geometry (101) of Buekenhout. We also give a detailed construction of this geometry. Moreover, we prove that, ifL3 (4) or L3(4) : 21are automorphism groups, this geometry is determined up to isomorphism by its diagram
Lifting Graph Automorphisms by Voltage Assignments

AbstractThe problem of lifting graph automorphisms along covering projections and the analysis of lifted groups is considered in a purely combinatorial setting. The main tools employed are: (1) a systematic use of the fundamental groupoid; (2) unification of ordinary, relative and permutation voltage constructions into the concept of a voltage space; (3) various kinds of invariance of voltage spaces relative to automorphism groups; and (4) investigation of geometry of the lifted actions by means of transversals over a localization set. Some applications of these results to regular maps on surfaces are given. Because of certain natural applications and greater generality, graphs are allowed to have semiedges. This requires careful re-examination of the whole subject and at the same time leads to simplification and generalization of several known results
On Score Sequences ofk-Hypertournaments

AbstractGiven two nonnegative integers n and k withn≥k> 1, a k -hypertournament on n vertices is a pair (V, A), where V is a set of vertices with | V | =n and A is a set of k -tuples of vertices, called arcs, such that for any k -subset S ofV , A contains exactly one of the k!k -tuples whose entries belong to S. We show that a nondecreasing sequence (r1, r2,⋯ , rn) of nonnegative integers is a losing score sequence of a k -hypertournament if and only if for each j(1 ≤j≤n),with equality holding whenj=n. We also show that a nondecreasing sequence (s1,s2 ,⋯ , sn) of nonnegative integers is a score sequence of somek -hypertournament if and only if for each j(1 ≤j≤n),with equality holding whenj=n. Furthermore, we obtain a necessary and sufficient condition for a score sequence of a strong k -hypertournament. The above results generalize the corresponding theorems on tournaments
Generalized Hook and Content Numbers Identities

AbstractGiven a cell x in a skew diagram, the arm length a(x), the leg length ℓ(x), and therefore the ‘hook pair’ (a(x),ℓ (x)), are defined. The diagram thus determines the corresponding multiset of its hook pairs. Similarly, x∈Z+2determines its ‘content pair’ (a′(x),ℓ′ (x)), which give raise to another type of multiset. Here we prove various identities between such multisets. In the special case of the hook and content numbers, these identities imply new ‘hook’ formulas for the dimensions of irreducible GL(n) representations. In the more general situation of the Jack symmetric functions Pλ(α), these identities imply new ‘α -hook’ formulas forPλ(α) (1, . . . , 1)
The Graphical Regular Representations of Finite Metacyclicp-Groups

AbstractA Cayley graph Γ=Cay(G, S) is called a graphical regular representation of the group G if AutΓ=G. One long-standing open problem about Cayley graphs is to determine which Cayley graphs are graphical regular representations of the corresponding groups. A simple necessary condition for Γ to be a graphical regular representation of G isAut (G, S) = 1, where Aut(G, S) = { τ∈Aut(G) |Sτ=S }. C. Godsil in (Europ. J. Combinatorics, 4 (1983)) proposed to characterize graphical regular representations of groups G in terms ofAut (G, S); that is, for a given class of groups G, find the conditions under whichCay (G, S) is a graphical regular representation of G if and only ifAut (G, S) = 1. The main purpose of this paper is to give a complete solution to this problem for the class of metacyclic p -groups where p is a prime
Almost 2-homogeneous Bipartite Distance-regular Graphs

AbstractLet Γ= (X, R) denote a bipartite distance-regular graph with diameter d≥ 4, and fix a vertex x of Γ. The Terwilliger algebra of Γ with respect to x is the subalgebra T ofMatX (C) generated by A, E*0,E*1,⋯ , E*d, where A is the adjacency matrix ofΓ , and where E*i denotes the projection onto the i th subconstituent ofΓ with respect to x. Let W denote an irreducible T -module. W is said to be thin whenever dim E*iW≤ 1 (0 ≤i≤d). The endpoint of W is min{ i |E*iW≠= 0}. It is known that a thin irreducibleT -module of endpoint 2 has dimension d− 3, d− 2, ord− 1. Γ is said to be 2-homogeneous whenever for all i(1 ≤i≤d− 1 ) and for all x, y,z∈X with ∂(x, y) = 2,∂ (x, z) =i, ∂(y,z ) =i, the number | Γ1(x) ∩Γ1(y) ∩Γi−1(z)| is independent ofx , y, z. Nomura has classified the 2-homogeneous bipartite distance-regular graphs. In this paper we study a slightly weaker condition. Γ is said to be almost 2-homogeneous whenever for all i(1 ≤i≤d− 2 ) and for all x, y,z∈X with ∂(x, y) = 2,∂ (x, z) =i, ∂(y,z ) =i, the number | Γ1(x) ∩Γ1(y) ∩Γi−1(z)| is independent ofx , y, z. We prove that the following are equivalent: (i)Γ is almost 2-homogeneous; (ii)Γ has, up to isomorphism, a unique irreducible T -module of endpoint 2 and this module is thin. Moreover, Γ is 2-homogeneous if and only if (i) and (ii) hold and the unique irreducible T -module of endpoint 2 has dimension d− 3
EveryH-decomposition ofKnhas a Nearly Resolvable Alternative

AbstractLet H be a fixed graph. An H -decomposition of Knis a coloring of the edges of Knsuch that every color class forms a copy of H. Each copy is called a member of the decomposition. The resolution number of an H -decomposition L of Kn, denoted χ(L), is the minimum number t such that the color classes (i.e., the members) of L can be partitioned into t subsets L1, . . . ,Lt , where any two members belonging to the same subset are vertex-disjoint. A trivial lower bound isχ (L) ≥n− 1__ dwhere __ d is the average degree of H. We prove that whenever Knhas an H -decomposition, it also has a decomposition L satisfyingχ (L) =n− 1__ d(1 +on(1))
Cartesian Products of Graphs and Metric Spaces

AbstractWe prove uniqueness of decomposition of a finite metric space into a product of metric spaces for a wide class of product operations. In particular, this gives the positive answer to the long-standing question of S. Ulam: ‘IfU×U≃V×V with U,V compact metric spaces, will then U and V be isometric?’ in the case of finite metric spaces. In the proof we use uniqueness of cartesian decomposition of connected graphs; a known fact to which we give a new proof which is shorter and more transparent than existing ones
On the Structure of 3-connected Matroids and Graphs

AbstractAn element e of a 3 -connected matroid M is essential if neither the deletionM\e nor the contraction M/e is 3 -connected. Tutte’s Wheels and Whirls Theorem proves that the only 3 -connected matroids in which every element is essential are the wheels and whirls. In this paper, we consider those 3 -connected matroids that have some non-essential elements, showing that every such matroid M must have at least two such elements. We prove that the essential elements of M can be partitioned into classes where two elements are in the same class if M has a fan, a maximal partial wheel, containing both. We also prove that if an essential element e of M is in more than one fan, then that fan has three or five elements; in the latter case, e is in exactly three fans. Moreover, we show that if M has a fan with 2 k or 2 k+ 1 elements for some k≥ 2, then M can be obtained by sticking together a (k+ 1)-spoked wheel and a certain 3 -connected minor of M. The results proved here will be used elsewhere to completely determine all 3 -connected matroids with exactly two non-essential elements