pinecone/core-2020-05-10-deduplication huggingface dataset:
Euler Cycles inK2mPlus Perfect Matching

AbstractWe1999 Academic Pressanalyze the freedom one has when constructing an Euler cycle throughCopyright K2m+, the complete graph of even order plus a perfect matching. We start with the fact that a multigraphGwithmedges is Eulerian iff it can be obtained from a cycle of lengthmby glueing certain vertices together. Our extension is the following. We deal with cycles of (2m2)+medges, whose vertices are colored black and white. There are 2mblack vertices and (2m2)−mwhite vertices. Assuming that glueing a black and a white vertex results in a black vertex we describe a method of how to glue certain vertices of a given cycle in such a way that the resulting multigraph isK2m+with all vertices black
Characterization of the Group Association Scheme of the Symmetric Group

AbstractLetnbe a non-zero positive integer and Λ(n) the set of all partitions ofn. There is a one-to-one correspondence between Λ(n) and the set of the conjugacy classes ofSn, the symmetric group of degreen. Let X(Sn)= (Sn, {R*λ}λ∈Λ(n)) be the group association scheme ofSnand X=(X, {Rλ}λ∈Λ(n)) be an association scheme having intersection numbers identical to those of X(Sn). Suppose there exists no set of four vertices {x1,x2,x3,x4} withx1,x2,x3,x4∈Xsatisfying (x1,x2), (x2,x3), (x3,x4), (x4,x1) ∈R(2), (x1,x3) ∈R(3)and (x2,x4) ∈R(2,2). Then X is shown to be isomorphic to X (Sn). (In [17], the authors show that ifn≥ 5, X does not possess four vertices of this type.
Exceptional Extended Dual Polar Spaces

AbstractWe address the classification problem of flag-transitive geometries with diagrams of the form where the leftmost edge symbolizes the geometry of vertices and edges of a complete graph ons+2 vertices and the residue of an element of the leftmost type is a finite thick classical dual polar space. These geometries are known asextended dual polar spaces. An extended dual polar space is called affine if it possesses a flag-transitive automorphism group which contains a normal subgroup acting regularly on the set of elements of the leftmost type. For a dual polar space D with three points per line there exists a unique 2-simply connected affine extension A(D) of D. We show that a flag-transitive extended dual polar space is either a quotient of A(D) for some D or isomorphic to one of 19 exceptional geometries whose full automorphism groups are isomorphic respectively to Sym8,U4(2).2,Sp6(2)×2,Sp6(2), 3·U4(3).22,U4(3).22,U5(2).2,McL.2,HS.2,Suz.2,Sp8(2), 3·Fi22.2,Fi22.2,Co2× 2,Co2,Fi24(s=4,t=2),Fi24(s=t=3),F1andFi23
Intersection Properties of Subsets of Integers

AbstractLetNkbe the maximal integer such that there exist subsetsA1,. . . ,ANk⊆ {1, 2,⋯ ,n} for whichAi∩Ajis an arithmetic progression of length at leastkfor every 1 ≤i<j≤Nk. Graham, Simonovits and Sós gave the exact value ofN0. Fork≥ 2, Simonovits and Sós determined the asymptotic behavior ofNk. In this paper we prove a conjecture of Simonovits and Sós concerning the asymptotic value ofN1. We show thatN1=n22+O(n5/3log3n).Moreover, we slightly improve the best-known construction, thus disproving their conjecture on the exact extremal system
UnLattices, ConstructionB, and AGM Iterations

AbstractThe quadratic and cubic arithmetic geometric means (AGMs) are known to be parametrized by Jacobian and bidimensional theta series respectively. We suggest an approach based on codes of lengthLover an alphabet of sizesto parametrize the general AGM of orderLand parameters. This is possible given the existence of number fields and quaternion algebras equipped with suitable quadratic forms. Expressions for theta series of lattices over the Gauss, Eisenstein and Hurwitz integers using Hamming weight enumerators of, respectively, binary, ternary and quaternary codes are derived
Acyclicity of Switching Classes

AbstractFor a finite undirected graphG= (V,E) and a subsetA⊆V, the vertex switching ofGbyAis defined as the graphGA= (V,E'), which is obtained fromGby removing all edges betweenAand its complementAand adding as edges all nonedges betweenAandA. The switching class [G] determined byGconsists of all vertex switchingsGAfor subsetsA⊆V. We prove that the trees of a switching class [G] are isomorphic to each other. We also determine the types of treesTthat have isomorphic copies in [G]. Finally we show that apart from one exceptional type of forest, the real forests in a switching class are isomorphic. Here a forest is real, if it is disconnected
Permutation Groups, Vertex-transitive Digraphs and Semiregular Automorphisms

AbstractA nonidentity element of a permutation group is said to be semiregular if all of its orbits have the same length. The work in this paper is linked to [6] where the problem of existence of semiregular automorphisms in vertex-transitive digraphs was posed. It was observed there that every vertex-transitive digraph of orderpkormp, wherepis a prime,k≥1 andm≤pare positive integers, has a semiregular automorphism. On the other hand, there are transitive permutation groups without semiregular elements [4]. In this paper, it is proved that every cubic vertex-transitive graph contains a semiregular automorphism, and moreover, it is shown that every vertex-transitive digraph of order 2p2, wherepis a prime, contains a semiregular automorphism
Minimal Null Designs and a Density Theorem of Posets

AbstractClassically,null designswere defined on the poset of subsets of a given finite set (boolean algebra). A null design is defined as a collection of weightedk-subsets such that the sum of the weights ofk-subsets containing at-subset is 0 for everyt-subset, where 0≤t<k≤n. Null designs are useful to understand designs or to construct new designs from a known one. They also deserve research as pure combinatorial objects. In particular, people have been interested in the minimum number ofk-subsets of non-zero weight to make a non-zero null design, and the characterization of the null designs with the minimal number ofk-subsets of non-zero weight, which we callminimalnull designs. Minimal null designs were used to construct explicit bases of the space of null designs.The definition of null designs can be extended to any poset which has graded structure (rankedposet) as the boolean algebra does. In this paper, we prove general theorems on the structure of the null designs of finite ranked posets, which also yield a density theorem of finite ranked posets. We apply the theorems to two special posets—the boolean algebra and the generalized (q-analogue of) boolean algebra—to characterize the minimal nullt-designs
On the Non-existence of Thas Maximal Arcs in Odd Order Projective Planes

AbstractIn this paper it is shown that given a non-degenerate elliptic quadric in the projective spacePG(2n−1,q),qodd, then there does not exist a spread ofPG(2n−1,q) such that each element of the spread meets the quadric in a maximal totally singular subspace. An immediate consequence is that the construction of [9’ does not give maximal arcs in projective planes forqodd. It is also shown that the all one vector is not contained in the binary code spanned by the tangents to an elliptic quadric inPG(3,q),qodd
Maps and Half-transitive Graphs of Valency 4

AbstractA subgroupG of automorphisms of a graphX is said to be 12- transitive if it is vertex- and edge- but not arc-transitive. The graphX is said to be 12-transitive if AutX is 12-transitive. The correspondence between regular maps and 12-transitive group actions on graphs of valency 4 is studied via the well known concept of medial graphs. Among others it is proved that under certain general conditions imposed on a map, its medial graph must be a 12-transitive graph of valency 4 and, vice versa, under certain conditions imposed on the vertex stabilizer, a 12-transitive graph of valency 4 gives rise to\break an irreflexible regular map. This way infinite families of 12-transitive graphs are constructed from known examples of regular maps. Conversely, known constructions of 12-transitive graphs of valency 4 give rise to new examples of irreflexible regular maps. In the end, the concept of a symmetric genus of a 12-transitive graph of valency 4 is introduced. In particular, 12-transitive graphs of valency 4 and small symmetric genuses are discussed
Generalized Perturbed Symmetry

AbstractLetwRdenote the reversal of the wordw, and letwE=w. We examine a generalization of Mendès France's ‘perturbed symmetry’. Specifically, we study a map of the formS(w) = w∏1≤ i≤ k−1xiwei,whereei ∈{E,R}, and the infinite word arising from iteration of this map. In particular, if thexiare fixed strings of identical length, the resulting infinite word isk-automatic, and we characterize precisely when such a word can be ultimately periodic. Finally, we apply our technique to prove a 1994 conjecture by Blanchard and Fabre
Double-(2n+1) Configurations in PG(2n+1, 2)

AbstractIn this paper we establish the existence of a configuration in PG(2n+1,2),n≥2, a particular case of which is described in detail in [3]. The general configuration consists of two sets of 2n+1 spaces of dimensionnrelated by a bijection so that two relatedn-spaces meet in an (n−1)-space, two unrelatedn-spaces from different sets meet in a point and distinctn-spaces from the same set are disjoint. We determine the full group of linear automorphisms of these configurations and show that, for eachn≥3, those in PG(2n+1,2) are not all projectively equivalent to one another
Digraphical Regular Representations of Infinite Finitely Generated Groups

AbstractA directed Cayley graphXis called a digraphical regular representation (DRR) of a groupGif the automorphism group ofXacts regularly onX. LetSbe a finite generating set of the infinite cyclic groupZ. We show that a directed Cayley graphX(Z,S) is aDRRofZif and only ifS≠S−1. IfX(Z,S) is not aDRRwe show thatAut(X(Z,S)) =D∞. As a general result we prove that a Cayley graphXof a finitely generated torsion-free nilpotent groupNis aDRRif and only if no non-trivial automorphism ofNof finite order leaves the generating set invariant
Partition Regular Inequalities

AbstractAnm×nrational matrixAis said to be partition regular if for every finite colouring of N there is a monochromatic vectorxwithAx=0. Given a partition regular matrixA, and a linear inequality such as ∑bixi>0, when is it true that for every finite colouring of N there is a monochromatic vectorxsatisfyingAx=0 and the additional constraint ∑bixi>0? This is certainly the case if we can guarantee that ∑bixi(or some fixed positive multiple of ∑bixi) is not merely positive but is actually the same colour as thexi. Our main aim in this paper is to show that this sufficient condition is also necessary. We give a similar result when there is more than one constraint. As a special case, we determine which matricesBhave the following property: for every finite colouring of N there is a monochromatic vectorxwith all entries ofBxpositive
Isotropic Vectors of Non-negative Integral Quadratic Forms

AbstractWe say that an integral quadratic formq:Zn→Z of the shapeq(x) = ∑ni=1qix(i)2+∑i<jqijx(i) × (j)is a semi-unit form ifqi≤1 for every 1≤i≤n. A deflation forqis the linear transformationT−ij:Zn→:Zndefined byes↦esifs≠jandej↦ej−eiifqij<0. We show that for a non-negative semi-unit formq:Zn→Z with corankq= corank+q, there exists an iterated sequence of deflationsTsuch thatq̄=qTsatisfiesq̄ij≥0 for alli,j. Further we describe the lattice of positive isotropic vectors of the non-negative semi-unit formsqwith corankq= corank+q=2
Cyclic Eulerian Elements

AbstractLetSnbe the symmetric group on {1,…,n} and Q[Sn] its group algebra over the rational field; we assumen≥2. π∈Snis said a descent ini, 1≤i≤n-1, if π(i)>π (i+1); moreover, π is said to have a cyclic descent if π(n)>π(1). We define the cyclic Eulerian elements as the sums of all elements inSnhaving a fixed global number of descents, possibly including the cyclic one. We show that the cyclic Eulerian elements linearly span a commutative semisimple algebra of Q[Sn], which is naturally isomorphic to the algebra of the classical Eulerian elements. Moreover, we give a complete set of orthogonal idempotents for such algebra, which are strictly related to the usual Eulerian idempotents
Jacobi–Trudi Identities for Boolean Tableaux and Ideal-tableaux of Zigzag Posets

AbstractAboolean tableauis an arrayT=(Tij) of the elements of a finite boolean algebra with several rows and infinitely many columns, where the entries increase from left to right and downwards. We study the generating functions for various classes of boolean tableaux. Applying the Gessel–Viennot method to certain nonplanar digraphs, we have determinantal formulas for the generating functions, which are regarded as generalized Jacobi–Trudi identities. By this theorem, we can also deal withideal-tableaux of zigzags, and give some new totally positive matrices
The Spectra of Complementary Subgraphs in a Strongly Regular Graph

AbstractAs an application of Jacobi's identity on complementary minors, one can exhibit a simple explicit relation between the characteristic polynomials of any pair of complementary induced subgraphs in a strongly regular graph. In particular, there is an explicit relation between the spectra of the first and second subconstituents with respect to any vertex. Several consequences will be presented; for example, there follow rather strong parametric restrictions on an SRG that has some second subconstituent isomorphic to an antipodal distance-regular graph of diameter three. The latter class of graphs also admits a palatable complementary-minors formula; this will be stated, together with an application
Constructingn-Manifolds from Spines

AbstractIn this paper, we describe a standard algorithm, which extends the ‘bijoin’ construction, introduced in [1] for the orientable case. It allows us to obtain all (orientable and non-orientable) closedn-manifolds from a suitable class of (n−1)-dimensional spines.© 1998 Academic Press\vskip\baselineskip In questo lavoro si descrive un algoritmo che generalizza la costruzione del ‘bijoin’ introdotto in [1] per il caso orientabile. Tale algoritmo permette di costruire tutte len-varietà, orientabili e non orientabili, a partire da opportune spine di dimensionen−1
A Convexity Lemma and Expansion Procedures for Bipartite Graphs

AbstractA hierarchy of classes of graphs is proposed which includes hypercubes, acyclic cubical complexes, median graphs, almost-median graphs, semi-median graphs and partial cubes. Structural properties of these classes are derived and used for the characterization of these classes by expansion procedures, for a characterization of semi-median graphs by metrically defined relations on the edge set of a graph and for a characterization of median graphs by forbidden subgraphs. Moreover, a convexity lemma is proved and used to derive a simple algorithm of complexityO(mn) for recognizing median graphs
Permutation Trees and Variation Statistics

AbstractIn this paper we exploit binary tree representations of permutations to give a combinatorial proof of Purtill's result [8] that∑δ∈Anvcd(δ)‖c=a+bd=ab+ba=∑σ∈Snvab(σ),where Anis the set of André permutations,vcd(σ) is thecd-statistic of an André permutation andvab(σ) is theab-statistic of a permutation. Using Purtill's proof as a motivation we introduce a new ‘Foata–Strehl-like’ action on permutations. This Z2n−1-action allows us to give an elementary proof of Purtill's theorem, and a bijection between André permutations of the first kind and alternating permutations starting with a descent. A modified version of our group action leads to a new class of André-like permutations with structure similar to that of simsun permutations
On Normal Cayley Graphs and Hom-idempotent Graphs

AbstractA graphGis said to behom-idempotentif there is a homomorphism fromG2toG, andweakly hom-idempotentif for somen ≥ 1 there is a homomorphism fromGn+ 1toGn. We characterize both classes of graphs in terms of a special class of Cayley graphs callednormal Cayley graphs. This allows us to construct, for any integern, a Cayley graphGsuch thatGn+1→Gn↛Gn−1, answering a question of Hahn, Hell and Poljak [8]. Also, we show that the Kneser graphs are not weakly hom-idempotent, generalizing a result of Albertson and Collins [1] for the Petersen graph
Flat Lax and Weak Lax Embeddings of Finite Generalized Hexagons

AbstractIn this paper we study laxly embedded generalized hexagons in finite projective spaces (a generalized hexagon is laxly embedded inPG(d,q) if it is a spanning subgeometry of the natural point-line geometry associated toPG(d,q)), satisfying the following additional assumption: for any pointxof the hexagon, the set of points collinear in the hexagon withxis contained in some plane ofPG(d,q). In particular, we show thatd≤7, and ifd=7, we completely classify all such embeddings. A classification is also carried out ford=5, 6 under some additional hypotheses. Finally, laxly embedded generalized hexagons satisfying other additional assumptions are considered, and classifications are also obtained
The Universal Covers of a Family of Extended Generalized Quadrangles

AbstractA sufficient condition for the simple connectedness is given for an infinite family of extended generalized quadrangles Y(S) of order (q+1,q−1) constructed in [7] from a family S of planes inPG(5,q) with some conditions. Applying this, Y(S) is shown to be simply connected when S is obtained from a (q+1)-arc inPG(3,q) except forq=4, and when S is constructed from the hyperovals for which the associated permutation polynomials are explicitly given in the list [6, p.299]}, except possibly for a class of Payne
On Isomorphisms of Finite Cayley Graphs

AbstractA Cayley graph Cay(G,S) of a groupGis called a CI-graph if wheneverTis another subset ofGfor which Cay(G,S)≅ Cay(G,T), there exists an automorphism σ ofGsuch thatS^σ= T. For a positive integerm, the groupGis said to have them-CI property if all Cayley graphs ofGof valencymare CI-graphs; further, ifGhas thek-CI property for allk≤m, thenGis called anm-CI-group, and a |G|-CI-groupGis called a CI-group. In this paper, we prove that Ais not a 5-CI-group, that SL(2,5) is not a 6-CI-group, and that all finite 6-CI-groups are soluble. Then we show that a nonabelian simple group has the 4-CI property if and only if it is A5, and that no nonabelian simple group has the 5-CI property. Also we give nine new examples of CI-groups of small order, which were found to be CI-groups with the assistance of a computer
Relation Algebras andt-vertex Condition Graphs

AbstractThe scheme associated with a graph is an association scheme iff the graph is strongly regular. Consider the problem of extending such an association scheme to a superscheme. The obstacles can be expressed in terms oft-vertex conditions. If a graph does not satisfy thet-vertex condition, a presuperscheme associated with it cannot be erected beyond the (t−3)rd level. We give an example of an association scheme which is not extendible to a superscheme: it cannot be extended beyond the bottom level of a presuperscheme
Hyperbolic Fibrations ofPG(3,q)

AbstractA hyperbolic fibration is set ofq−1 hyperbolic quadrics and two lines which together partition the points ofPG(3,q). The classical example of a hyperbolic fibration comes from a pencil of quadrics; however, several other families are known. In this paper we construct a new family of hyperbolic fibrations for odd prime powersq.As an application of hyperbolic fibrations, we note that they can be used to construct 2q−1(not necessarily inequivalent) spreads ofPG(3,q) by choosing one ruling family from each of the hyperbolic quadrics in the fibration. For our new fibration we discuss some properties of the spreads obtained in the above manner
Generalised Classes in Groups and Association Schemes: Duals of Results on Characters and Sharpness

AbstractA result of Strunkov on generalised conjugacy classes of groups is most conveniently expressed in terms of the P-matrix of an association scheme. This result is dual to a result of Blichfeldt on permutation characters, which has been shown by Cameron and Kiyota to hold for arbitrary characters and led to the definition of sharp characters. We show that the result carries over to arbitrary (commutative) association scheme classes, and introduce the idea of a sharp class in an association scheme
Upper Bounds of Entire Chromatic Number of Plane Graphs

AbstractThe1999 Academic Pressentire chromatic number χCopyright vef(G) of a plane graphGis the least number of colors assigned to the vertices, edges and faces so that every two adjacent or incident pair of them receive different colors. Kronk and Mitchem (1973) conjectured that χvef(G)≤ Δ+4 for every plane graphG. In this paper we prove the conjecture for a plane graphGhaving χ′(G)=Δ and give a upper bound χvef(G) ≤ Δ+5 for all plane graphs, where χ′(G) and Δ are the chromatic index and the maximum degree ofG, respectively
On Lih's Conjecture concerning Spernerity

AbstractLet F be a nonempty collection of subsets of [n] ={ 1, 2, …,n}, each having cardinalityt. Denote byPFthe poset consisting of all subsets of [n] which contain at least one member of F, ordered by set-theoretic inclusion. In 1980, K. W. Lih conjectured thatPFhas the Sperner property for all 1 ≤ t≤nand every choice of F. This conjecture is known to be true fort=1 but false, in general, fort≥ 4. In this paper, we prove Lih's conjecture in the caset=2.We make extensive use of fundamental theorems concerning the preservation of Sperner-type properties under direct products of posets
On a Conjecture of Cameron and Liebler

AbstractCameron–Liebler line classes arose from an attempt by Cameron and Liebler to classify those collineation groups ofPG(n,q) which have the same number of orbits on points as on lines. They satisfy several equivalent properties; among them, constant intersection with spreads. Cameron and Liebler conjectured that, apart from some ‘obvious’ examples, no sets of lines of this type exist inPG(3,q). This paper introduces a connection between Cameron–Liebler line classes inPG(3,q) and blocking sets inPG(2,q), and uses it to provide the strongest results to date concerning the non-existence of certain of these sets. In addition, a complete classification of Cameron–Liebler line classes inPG(3,3) is obtained, with the main result being that there is, essentially, a unique counterexample to Cameron and Liebler's conjecture in this space
Judicious Partitions of 3-uniform Hypergraphs

AbstractA conjecture of Bollobás and Thomason asserts that, for r≥ 1, everyr -uniform hypergraph with m edges can be partitioned into r classes such that every class meets at least rm/(2 r− 1) edges. Bollobás, Reed and Thomason proved that there is a partition in which every edge meets at least (1 − 1/e)m/3 ≈ 0.21 m edges. Our main aim is to improve this result for r= 3. We prove that every 3-uniform hypergraph with m edges can be partitioned into three classes, each of which meets at least (5 m− 1)/9 edges. We also prove that for r> 3 we may demand 0.27 m edges
Affine Distance-transitive Graphs with Sporadic Stabilizer

AbstractThis paper is a contribution to the programme to classify finite distance-transitive graphs and their automorphism groups. We classify pairs (Γ,G) where Γ is a graph andGis an automorphism group of Γ acting distance-transitively and primitively on the vertex set of Γ, subject to the condition that there is a normal elementary abelian subgroupVinGwhich acts regularly on the vertex set of Γ and the stabilizerG0of a vertex (which is a complement toVinG) has a unique non-abelian composition factor isomorphic to one of the 26 sporadic simple groups. There are exactly 10 examples of Γ, all known for a long time