pinecone/core-2020-05-10-deduplication huggingface dataset:
Asymptotic Clique Covering Ratios of Distance Graphs

AbstractGiven a finite set D of positive integers, the distance graph G(Z, D) has Z as the vertex set and { ij: | i−j | ∈D } as the edge set. Given D, the asymptotic clique covering ratio is defined as S(D) =lim supn→∞ncl(n), where cl(n) is the minimum number of cliques covering any consecutive n vertices of G(Z, D). The parameter S(D) is closely related to the ratio spT(G)χ(G)of a graph G, where χ(G) and spT(G) denote, respectively, the chromatic number and the optimal span of a T -coloring of G. We prove that for any finite set D, S(D) is a rational number and can be realized by a ‘periodical’ clique covering of G(Z, D). Then we investigate the problem for which sets D the equality S(D) =ω(G(Z, D)) holds. (In general, S(D) ≤ω(G(Z, D)), where ω(G) is the clique number of G.) This problem turns out to be related to T -colorings and to fractional chromatic number and circular chromatic number of distance graphs. Through such connections, we shall show that the equality S(D) =ω(G(Z, D)) holds for many classes of distance graphs. Moreover, we raise questions regarding other such connections
Spreads in H(q) and 1-systems of Q(6,q )

AbstractIn this paper we prove that the projections along reguli of a translation spread of the classical generalized hexagon H(q) are translation ovoids of Q(4, q). As translation ovoids of Q(4, 2r) are elliptic quadrics, this forces that all translation spreads ofH (2r) are semi-classical. By representing H(q) as a coset geometry, we obtain a characterization of a translation spread in terms of a set of points of PG(3, q) which belong to imaginary chords of a twisted cubic and we construct a new example of a semi-classical spread ofH (2r). Finally, we study the semi-classical locally Hermitian 1-systems ofQ (6, q) which are spreads of Q−(5, q)
On the Bi-embeddability of Certain Steiner Triple Systems of Order 15

AbstractThere are 80 non-isomorphic Steiner triple systems of order 15. A standard listing of these is given in Mathon et al.(1983, Ars Combin., 15, 3–110). We prove that systems #1 and #2 have no bi-embedding together in an orientable surface. This is the first known example of a pair of Steiner triple systems of ordern , satisfying the admissibility condition n≡ 3 or 7(mod 12), which admits no orientable bi-embedding. We also show that the same pair has five non-isomorphic bi-embeddings in a non-orientable surface
Continued Fractions and Generalized Patterns

AbstractBabson and Steingrimsson (2000, Séminaire Lotharingien de Combinatoire, B44b, 18) introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation.Let fτ;r(n) be the number of 1 - 3 - 2-avoiding permutations on n letters that contain exactly r occurrences of τ, where τ is a generalized pattern on k letters. Let Fτ;r(x) and Fτ(x, y) be the generating functions defined by Fτ;r(x) =∑ n≥0fτ;r(n)xnand Fτ(x, y) =∑ r≥0Fτ;r(x)yr. We find an explicit expression for Fτ(x, y) in the form of a continued fraction for τ given as a generalized pattern: τ= 12 - 3 -⋯-k, τ= 21 - 3 -⋯-k, τ= 123⋯k, or τ=k⋯321. In particular, we find Fτ(x, y) for any τ generalized pattern of length 3. This allows us to express Fτ;r(x) via Chebyshev polynomials of the second kind and continued fractions
Regular Homomorphisms and Regular Maps

AbstractRegular homomorphisms of oriented maps essentially arise from a factorization by a subgroup of automorphisms. This kind of map homomorphism is studied in detail, and generalized to the case when the induced homomorphism of the underlying graphs is not valency preserving. Reconstruction is treated by means of voltage assignments on angles, a natural extension of the common assignments on darts. Lifting and projecting groups of automorphisms along regular homomorphisms is studied in some detail. Finally, the split-extension structure of lifted groups is analysed
Hamiltonian Tournaments and Gorenstein Rings

AbstractLet Gnbe the complete graph on the vertex set [ n ] = {1, 2,⋯ , n } and ω an orientation of Gn, i.e.,ω is an assignment of a direction i→j of each edge { i, j } of Gn. Let eqdenote the q th unit coordinate vector of Rn. WriteP(Gn;ω)⊂Rn for the convex hull of the ( n 2) pointsei−ej, wherei→j is the direction of the edge { i, j } in the orientationω . It will be proved that, for n≥ 5, the Ehrhart ring of the convex polytopeP(Gn;ω) is Gorenstein if and only if (Gn; ω) possesses a Hamiltonian cycle, i.e., a directed cycle of lengthn
A Common Extension of the Erdős–Stone Theorem and the Alon–Yuster Theorem for Unbounded Graphs

AbstractThe Erdős–Stone theorem (1946, Bull. Am. Math. Soc., 52, 1089–1091) and the Alon–Yuster theorem (1992, Graphs Comb., 8, 95–102 ) are both very fundamental in extremal graph theory. We give a common extension of them, which states as follows: For everyϵ> 0 and r≥ 2, there exists c=cϵ, r> 0 such that, for any 0 ≤θ≤ 1, if H is a graph of order | H | ≤clogn and with chromatic number r then every n -vertex graph G with minimum degree at least ( 1 − 1 __ r−1 +θ ____ r(r−1)) n contains at least (θ−ϵ)n/| H | vertex-disjoint copies of H.When θ=ϵr (r− 1) or θ= 1, it would imply the two theorems.The important point is that our theorem enables us to deal with a larger graph H of order | H | →∞(as n→∞), while | H | was fixed in the Alon–Yuster theorem (and in another common extension by Komlós (2000, Combinatorica,20, 203–218)).The bounds clogn and ( 1 − 1 __ r−1 +θ ____r (r−1)) n are both essentially the best possible
Almost-Spanning Subgraphs with Bounded Degree in Dense Graphs

AbstractWe present two extensions of a theorem by Alon and Yuster (1992, Graphs Comb., 8, 95–102) that give degree conditions guaranteeing an almost-spanning subgraph isomorphic to a given graph. The first extension gives a sharp degree condition when the desired subgraph consists of small connected components (i.e., a packing of a host graph with small graphs), improving a theorem of Komlós (2000, Combinatorica, 20, 203– 218). The second extension weakens the assumption of the desired subgraph in the Alon–Yuster theorem.Given a graph F, we write χ(F) and Δ(F) for the chromatic number and the maximum degree, respectively. We also denote by σr(F) the smallest possible colour class size in any proper r -vertex-colouring of F. The first theorem states that, for every Δ≥ 1 and ϵ> 0, there exists a μ> 0 and an n0such that the following holds. Let H= ∪iHibe a non-empty graph such that | H | ≤ (1 − ϵ)n, Δ(H) ≤ Δ, and, for eachHi, | Hi| ≤ μn.Then every graph G with order n≥n0and minimum degree δ(G) ≥ ( 1 − 1 − σr− 1 − μ)ncontains a copy of H where r:=maxiχ(Hi) and σ:=max{∑iσr(Hi)/| H | , ϵ }. The second theorem states that, for any r> 1, Δ≥ 0, and ϵ> 0, there exists a μ> 0 and an n0such that for every graph G with order n≥n0and δ(G) ≥ ( 1 − 1 r− μ)n,one of the following two holds: • For any graph H with | H | ≤ (1 − ϵ)n, Δ(H) ≤ Δ, χ(H) ≤ r, andb(H) ≤ μn,G contains a copy of H(where b(H) denotes the bandwidth of H). • By deleting and adding at most ϵn2edges and r vertices of G, G can be isomorphic to Kr(⌊n/r⌋) orK⌊n/r⌋+Kr−2(⌊n/r⌋) +K⌊n/r⌋. The assumption of b(H) cannot be dropped
On 1-Arc-regular Graphs

AbstractA graph is 1-arc-regular if its full automorphism group acts regularly on the set of its arcs. In this paper, we investigate 1-arc-regular graphs of prime valency, especially of valency 3. First, we prove that if G is a soluble group then a (G, 1)-arc-regular graph must be a Cayley graph of a subgroup of G. Next we consider trivalent Cayley graphs of a finite nonabelian simple group and obtain a sufficient condition under which one can guarantee that Cay(G, S) is 1-arc-regular. Finally, as an application of the result, we construct two infinite families of 1-arc-regular trivalent Cayley graphs with insoluble automorphism groups and, in particular, one of the families is not a Cayley graph
Bases in Orlik–Solomon Type Algebras

AbstractLet M be a matroid on [ n ] and E be the graded algebra generated over a field k generated by the elements 1, e1,⋯ , en. Let (M) be the ideal of E generated by the squares e12,⋯ , en2, elements of the form eiej+aijejeiand ‘boundaries of circuits’, i.e., elements of the form ∑χjei1⋯eij−1eij+1⋯eim, with χj∈k and ei1,⋯ , eima circuit of the matroid with some special coefficients. The χ -algebra A(M) is defined as the quotient of E by (M). Recall that the class of χ -algebras contains several studied algebras and in first place the Orlik–Solomon algebra of a matroid. We will essentially construct the reduced Gröbner basis of (M) for any term order and give some consequences
Some New Large Compound Graphs

AbstractThis paper deals with some new constructions of large (Δ, D) graphs, i.e., graphs with maximum degree Δ and diameter D and many vertices. Most constructions presented here are based on the compound graphs technique. The basic idea of compound graphs consists of connecting together several copies of a given graph according to the structure of another one.Using a variation on this classical theme, we build some new large graphs and some new large bipartite graphs. We will also apply the method of vertex duplication to improve some entries in the tables of known large bipartite graphs
On Line Arrangements in the Hyperbolic Plane

AbstractGiven a finite collection L of lines in the hyperbolic plane H, we denote byk=k(L) its Karzanov number, i.e., the maximal number of pairwise intersecting lines in L, and by C(L) and n=n (L) the set and the number, respectively, of those points at infinity that are incident with at least one line from L. By using purely combinatorial properties of cyclic sets, it is shown that#L≤ 2 nk− ( 2 k+1 2 ) always holds and that #L equals 2 nk− ( 2 k+1 2 ) if and only if there is no collection L′ of lines in H with L⊊L′,k (L′) =k(L) andC (L′) =C(L)
Valency of Distance-regular Antipodal Graphs with Diameter 4

AbstractLet G be a non-bipartite strongly regular graph on n vertices of valency k. We prove that if G has a distance-regular antipodal cover of diameter 4, then k≤ ⌊2(n+ 1)/5⌋, unless G is the complement of triangular graph T(7), the folded Johnson graph J(8, 4) or the folded halved 8-cube. However, for these three graphs the bound k≤ ⌊(n− 1)/2⌋ holds. This result implies that only one of a complementary pair of strongly regular graphs can be the antipodal quotient of an antipodal distance-regular graph
Three-fold Restricted Set Addition in Groups

AbstractLet G be a finite abelian group with a ‘sufficiently small’ proportion of elements of order two. We prove that for any subset A⊆G of cardinality | A | >513| G | and any group elementg∈G, there exist pairwise distinct a1, a2,a3∈A such that a1+a2+a3=g— unless A is contained in a coset of an index two subgroup or in a union of two cosets of an index five subgroup.As an immediate corollary our result implies a recent conjecture of Gallardo et al. who addressed the groupG=Znwith n odd and sets of cardinality | A | >25n
Bridging Semisymmetric and Half-Arc-Transitive Actions on Graphs

AbstractA generalization of some of Folkman’s constructions (see (1967) J. Comb. Theory, 3, 215–232) of the so-called semisymmetric graphs, that is regular graphs which are edge- but not vertex-transitive, was given by Marušič and Potočnik (2001, Europ. J. Combinatorics,22, 333–349) together with a natural connection between graphs admitting 1/2-arc-transitive group actions and certain graphs admitting semisymmetric group actions. This connection is studied in more detail in this paper. Among others, a sufficient condition for the semisymmetry of the so-called generalized Folkman graphs arising from certain graphs admitting a 1/2-arc-transitive group action is given. Furthermore, the concepts of alter-sequence and alter-exponent is introduced and studied in great detail and then used to study the interplay of three classes of graphs: cubic graphs admitting a one-regular group action, the corresponding line graphs which admit a 1/2-arc-transitive action of the same group and the associated generalized Folkman graphs. At the end an open problem is posed, suggesting an in-depth analysis of the structure of tetravalent 1/2-arc-transitive graphs with alter-exponent 2
Constructing an Infinite Family of Cubic 1-Regular Graphs

AbstractA graph is 1- regular if its automorphism group acts regularly on the set of its arcs. Miller [J. Comb. Theory, B , 10 (1971), 163–182] constructed an infinite family of cubic 1-regular graphs of order 2 p, where p≥ 13 is a prime congruent to 1 modulo 3. Marušič and Xu [ J. Graph Theory, 25 (1997), 133–138] found a relation between cubic 1-regular graphs and tetravalent half-transitive graphs with girth 3 and Alspach et al. [ J. Aust. Math. Soc. A, 56 (1994), 391–402] constructed infinitely many tetravalent half-transitive graphs with girth 3. Using these results, Miller’s construction can be generalized to an infinite family of cubic 1-regular graphs of order 2n , where n≥ 13 is odd such that 3 divides ϕ(n), the Euler function of n. In this paper, we construct an infinite family of cubic 1-regular graphs with order 8(k2+k+ 1)(k≥ 2) as cyclic-coverings of the three-dimensional Hypercube
The t-intersection Problem in the Truncated Boolean Lattice

AbstractLet I(n, t) be the class of all t -intersecting families of subsets of [ n ] and set Ik(n, t) =I (n, t) ∩ 2[ n ]k, I≤k(n, t) =I(n, t) ∩ 2[ n ] ≤k.After the maximal families inI (n, t) [13] and in Ik(n, t) [1,9] are known we study now maximal families in I≤k(n, t). We present a conjecture about the maximal cardinalities and prove it in several cases.More generally cardinalities are replaced by weights and asymptotic estimates are given.Analogous investigations are made for I(n,t ) ∩C(n, s), where C(n,s ) is the class of all s -cointersecting families of subsets of [ n ]. In particular we establish an asymptotic form of a conjecture by Bang et al. [4]
Hyperbolic Bridged Graphs

AbstractGiven a connected graph G, we take, as usual, the distance xy between any two verticesx , y of G to be the length of some geodesic between x and y. The graph G is said to be δ - hyperbolic, for some δ≥ 0, if for all vertices x,y , u, v in G the inequality xy+uv ≤max{ xu+ yv,xv + yu } + δholds, and G isbridged if it contains no finite isometric cycles of length four or more. In this paper, we will show that a finite connected bridged graph is 1-hyperbolic if and only if it does not contain any of a list of six graphs as an isometric subgraph
Full Embeddings of (α, β) -Geometries in Projective Spaces

AbstractThe incidence structures known as (α, β)-geometries are a generalization of partial geometries and semipartial geometries. For an (α, β)-geometry fully embedded in PG (n, q), the restriction to a plane turns out to be important. Planes containing an antiflag of the (α, β)-geometry can be divided into α -planes,β -planes and mixed planes. In this paper (1, β)-geometries fully embedded in PG (3,q ) are classified under the assumption that PG (3, q) contains at least one 1-plane and at least one β -plane. Next we classify (α, β)-geometries fully embedded in PG (n, q), for α> 1 and q odd, under the assumption that every plane of PG (n, q) that contains an antiflag of S is either an α -plane or aβ -plane . We also treat the case that there is a mixed plane and that β=q+ 1. In a forthcoming paper we will treat the case β=q . The cases β=q and β=q+ 1 are the only cases that can occur under the assumptions that q is odd,α> 1 and that there is at least one β -plane
Zero-sum Square Matrices

AbstractLet A be a matrix over the integers, and let p be a positive integer. A submatrix B of A is zero-summodp if the sum of each row of B and the sum of each column of B is a multiple of p. Let M(p, k) denote the least integer m for which every square matrix of order at least m has a square submatrix of order k which is zero-sum modp. In this paper we supply upper and lower bounds for M(p, k). In particular, we prove that limsupM(2, k)/k≤ 4, liminfM(3, k)/k≤ 20, and that M(p, k) ≥k22 eexp(1/e)p/2. Some nontrivial explicit values are also computed
Tight Distance-regular Graphs and the Subconstituent Algebra

AbstractWe consider a distance-regular graph Γ with diameter D≥ 3, intersection numbers ai, bi, ciand eigenvalues k=θ0>θ1>⋯>θD. Let X denote the vertex set of Γ and fix x∈X. Let T=T(x) denote the subalgebra of Mat X(C) generated by A, E0*, E1*,⋯ , ED*, where A denotes the adjacency matrix of Γ and Ei*denotes the projection onto the i th subconstituent of Γ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T -module W is said to be thin whenever dimEi*W≤ 1 for 0 ≤i≤D. By the endpoint of W we mean min{ i | Ei*W≠= 0}. Let W denote a thin irreducible T -module with endpoint 1. Observe E1*W is a one-dimensional eigenspace for E1*AE1*; let η denote the corresponding eigenvalue. We call η the local eigenvalue of W. It is known yeujc5970x.gif θ1≤η≤ yeujc5971x.gif θDwhere yeujc5972x.gif θ1=−1 −b1(1 +θ1)−1and yeujc5973x.gif θD=−1 −b1(1 +θD)−1. Let n= 1 or n=D and assume η= yeujc5974x.gif θn. We show the dimension of W is D− 1. Let v denote a nonzero vector in E1*W. We show W has a basis Eiv(1 ≤i≤D, i≠=n), where Eidenotes the primitive idempotent of A associated with θi. We show this basis is orthogonal (with respect to the Hermitean dot product) and we compute the square norm of each basis vector. We show W has a basis Ei+1*Aiv(0 ≤i≤D− 2), where Aidenotes the i th distance matrix for Γ. We find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square norm of each basis vector. We find the transition matrix relating our two bases for W. For notational convenience, we say Γ is 1- thin with respect tox whenever every irreducible T -module with endpoint 1 is thin. Similarly, we say Γ is tight with respect tox whenever every irreducible T -module with endpoint 1 is thin with local eigenvalue yeujc5975x.gif θ1or yeujc5976x.gif θD. In [ J. Algebr. Comb., 12,(2000), 163–197] Jurišić, Koolen and Terwilliger showed (θ1+ka1+ 1) (θD+ka1+ 1) ≥−ka1b1(a1+ 1)2.They defined Γ to be tight whenever Γ is nonbipartite and equality holds above. We show the following are equivalent: (i) Γ is tight; (ii) Γ is tight with respect to each vertex; (iii) Γ is tight with respect to at least one vertex. We show the following are equivalent: (i) Γ is tight; (ii) Γ is nonbipartite, aD= 0, and Γ is 1-thin with respect to each vertex; (iii) Γ is nonbipartite, aD= 0, and Γ is 1-thin with respect to at least one vertex
On a Particular Class of Minihypers and its Applications. III. Applications

AbstractIn the first two articles of this series, the structure of certain minihypers was determined. Hamada shows how these results translate into results on linear codes meeting the Griesmer bound, while Govaerts and Storme show how they can be applied to obtain bounds on the size of maximal partial t -spreads and minimal t -covers in finite projective spaces that admit a t -spread. In this article, further applications are given. It is shown that the previously studied minihypers are closely connected to partial t -spreads and t -covers of finite classical polar spaces whose size admits a t -spread. This connection is used to obtain new bounds on the sizes of maximal partial t -spreads of finite classical polar spaces whose sizes admit t -spreads. In order to get a clearer view on which polar spaces these are, divisibility conditions are rewritten into a more convenient form. This yields necessary conditions for the existence of t -spreads in those spaces; it turns out that for some of the polar spaces these conditions are also sufficient. The results on minihypers are then applied to t -covers of the classical polar spaces, and give us a better understanding of their structure. As an immediate corollary to an extendability result for partial t -spreads, a theorem on the extendability of partial ovoids of H(3, q2) is given. This theorem is then used to prove a new upper bound on the size of partial ovoids of H(4, q2), which can be lifted to an upper bound on the size of partial ovoids of H(2 n, q2),n≥ 2. Also partial ovoids on the generalized hexagon H(q) are studied
Constructing a Class of Symmetric Graphs

AbstractWe find a natural construction of a large class of symmetric graphs from point- and block-transitive 1-designs. The graphs in this class can be characterized as G -symmetric graphs whose vertex sets admit a G -invariant partition B of block size at least 3 such that, for any two blocks B, C of B, either there is no edge between B and C, or there exists only one vertex in B not adjacent to any vertex inC . The special case where the quotient graph ΓBof Γ relative to B is a complete graph occurs if and only if the 1-design needed in the construction is a G -doubly transitive and G -block-transitive 2-design, and in this case we give an explicit classification of Γ when G is a doubly transitive projective group or an affine group containing the affine general group. Examples of such graphs include cross ratio graphs studied recently by Gardiner, Praeger and Zhou and some other graphs with vertices the (point, line)-flags of the projective or affine geometry
Hamilton-connectivity of 3-Domination Critical Graphs with α=δ+ 2

AbstractA graph G is 3-domination critical if its domination number \gamma is 3 and the addition of any edge decreases \gamma by 1. It was proved by Favaron et al. that α≤δ+ 2 for any connected 3-domination critical graph. Denote byτ (G) the toughness of a graph G. Recently Chen et al. conjectured that a connected 3-domination critical graph G is Hamilton-connected if and only if τ(G) > 1 and showed the conjecture is true when α≤δ. In this paper, by using a closure operation defined by Bondy and Chvátal, we show the conjecture is true whenα=δ+ 2
On Greedy Bases Packing in Matroids

AbstractLet S be a finite set and M= (S, B) be a matroid where B is the set of its bases. We say that a basis B is greedy in M or the pair (M, B) is greedy if, for every sum of bases vector w, the coefficient:λ(B, w) =max{λ ≥ 0 : w− λBisagainasumofbasesvector }, where B and its characteristic vector will not be distinguished, is integer. We define a notion of minors for (M, B) pairs and we give a characterization of greedy pairs by excluded minors. This characterization gives a large class of matroids for which an integer Carathéodory’s theorem is true