Metric characterization of apartments in dual polar spaces

Let Π be a polar space of rank n and let Gk(Π), k∈{0,…,n−1} be the polar Grassmannian formed by k-dimensional singular subspaces of Π. The corresponding Grassmann graph will be denoted by Γk(Π). We consider the polar Grassmannian Gn−1(Π) formed by maximal singular subspaces of Π and show that the image of every isometric embedding of the n-dimensional hypercube graph Hn in Γn−1(Π) is an apartment of Gn−1(Π). This follows from a more general result concerning isometric embeddings of Hm, m?n in Γn−1(Π). As an application, we classify all isometric embeddings of Γn−1(Π) in Γn^′−1(Π^′), where Π^′ is a polar space of rank n^′?n.     

On the fundamental group-scheme

We show that the fundamental group-scheme of a separably rationally connected variety defined over an algebraically closed field is trivial. Let X be a geometrically irreducible smooth projective variety defined over a finite field k admitting a k-rational point. Let {En,σn}_n?0 be a flat principal G-bundle over X, where G is a reductive linear algebraic group defined over k. We show that there is a positive integer a such that the principal G-bundle is isomorphic to E₀, where FX is the absolute Frobenius morphism of X. From this it follows that E₀ is given by a representation of the fundamental group-scheme of X in G.     

Frobenius splitting and geometry of G-Schubert varieties

Let X be an equivariant embedding of a connected reductive group G over an algebraically closed field k of positive characteristic. Let B denote a Borel subgroup of G. A G-Schubert variety in X is a subvariety of the form diag(G)⋅V, where V is a B×B-orbit closure in X. In the case where X is the wonderful compactification of a group of adjoint type, the G-Schubert varieties are the closures of Lusztig's G-stable pieces. We prove that X admits a Frobenius splitting which is compatible with all G-Schubert varieties. Moreover, when X is smooth, projective and toroidal, then any G-Schubert variety in X admits a stable Frobenius splitting along an ample divisors. Although this indicates that G-Schubert varieties have nice singularities we present an example of a nonnormal G-Schubert variety in the wonderful compactification of a group of type G₂. Finally we also extend the Frobenius splitting results to the more general class of R-Schubert varieties.     

A note on internally disjoint alternating paths in bipartite graphs

Let G be a balanced bipartite graph with partite sets X and Y, which has a perfect matching, and let x∈X and y∈Y. Let k be a positive integer. Then we prove that if G has k internally disjoint alternating paths between x and y with respect to some perfect matching, then G has k internally disjoint alternating paths between x and y with respect to every perfect matching.     

p-Rank and semi-stable reduction of curves

Let R be a discrete complete valuation ring, with field of fractions K, and with algebraically closed residue field k of characteristic p > 0. Let X be a germ of an R-curve at an ordinary double point. Consider a finite Galois covering f: Y → X, whose Galois group G is a p-group, such that Y is normal, and which is étale above X_k≔ x × _rk. Asume that Y has a semi-stable model :→ Y over R, and let y be a closed point of Y. If the inertia subgroup I(y) at y is cyclic of order pⁿ, we compute the p-rank of tf⁻¹ (y) by using a result of Raynaud. In particular, we prove that this p-rank is bounded by pⁿ −1.     

Boundaries and JSJ Decompositions of CAT(0)-Groups

Let G be a one-ended group acting discretely and co-compactly on a CAT(0) space X. We show that ?X has no cut points and that one can detect splittings of G over two-ended groups and recover its JSJ decomposition from ?X. We show that any discrete action of a group G on a CAT(0) space X satisfies a convergence type property. This is used in the proof of the results above but it is also of independent interest. In particular, if G acts co-compactly on X, then one obtains as a corollary that if the Tits diameter of ?X is bigger than 3π/2 then it is infinite and G contains a free subgroup of rank 2.     

On 3-Steiner simplicial orderings

Let G be a connected graph and S a nonempty set of vertices of G. A Steiner tree for S is a connected subgraph of G containing S that has a minimum number of edges. The Steiner interval for S is the collection of all vertices in G that belong to some Steiner tree for S. Let k≥2 be an integer. A set X of vertices of G is k-Steiner convex if it contains the Steiner interval of every set of k vertices in X. A vertex x∈X is an extreme vertex of X if X?{x} is also k-Steiner convex. We call such vertices k-Steiner simplicial vertices. We characterize vertices that are 3-Steiner simplicial and give characterizations of two classes of graphs, namely the class of graphs for which every ordering produced by Lexicographic Breadth First Search is a 3-Steiner simplicial ordering and the class for which every ordering of every induced subgraph produced by Maximum Cardinality Search is a 3-Steiner simplicial ordering.     

Star complements and exceptional graphs

Let G be a finite graph of order n with an eigenvalue μ of multiplicity k. (Thus the μ-eigenspace of a (0,1)-adjacency matrix of G has dimension k.) A star complement for μ in G is an induced subgraph G-X of G such that |X|=k and G-X does not have μ as an eigenvalue. An exceptional graph is a connected graph, other than a generalized line graph, whose eigenvalues lie in [-2,∞). We establish some properties of star complements, and of eigenvectors, of exceptional graphs with least eigenvalue −2.     

On the locating chromatic number of Kneser graphs

Let c be a proper k-coloring of a connected graph G and Π=(C₁,C₂,…,C_k) be an ordered partition of V(G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple c_Π(v):=(d(v,C₁),d(v,C₂),…,d(v,C_k)), where d(v,C_i)=min{d(v,x)|x∈C_i},1≤i≤k. If distinct vertices have distinct color codes, then c is called a locating coloring. The minimum number of colors needed in a locating coloring of G is the locating chromatic number of G, denoted by χ_L(G). In this paper, we study the locating chromatic number of Kneser graphs. First, among some other results, we show that χ_L(KG(n,2))=n−1 for all n≥5. Then, we prove that χ_L(KG(n,k))≤n−1, when n≥k². Moreover, we present some bounds for the locating chromatic number of odd graphs.     

On degree sum conditions for long cycles and cycles through specified vertices

Let G be a graph. For S⊂V(G), let Δ_k(S) denote the maximum value of the degree sums of the subsets of S of order k. In this paper, we prove the following two results. (1) Let G be a 2-connected graph. If Δ₂(S)≥d for every independent set S of order κ(G)+1, then G has a cycle of length at least min{d,|V(G)|}. (2) Let G be a 2-connected graph and X a subset of V(G). If Δ₂(S)≥|V(G)| for every independent set S of order κ(X)+1 in G[X], then G has a cycle that includes every vertex of X. This suggests that the degree sum of nonadjacent two vertices is important for guaranteeing the existence of these cycles.     

Principal bundles whose restrictions to a curve are isomorphic

Let X be a normal projective variety defined over an algebraically closed field k. Let |O _X (1)| be a very ample invertible sheaf on X. Let G be an affine algebraic group defined over k. Let E _G and F _G be two principal G-bundles on X. Then there exists an integer n > > 0 (depending on E _G and F _G ) such that if the restrictions of E _G and F _G to a curve C ∈ |O _X (n)| are isomorphic, then they are isomorphic on all of X.     

Orbit-equivalent infinite permutation groups

Let G,H be closed permutation groups on an infinite set X, with H a subgroup of G. It is shown that if G and H are orbit-equivalent, that is, have the same orbits on the collection of finite subsets of X, and G is primitive but not 2-transitive, then G=H.     

Neighborhoods and Covering Vertices by Cycles

G =(V,E) is a 2-connected graph, and X is a set of vertices of G such that for every pair x,x' in X, , and the minimum degree of the induced graph <X> is at least 3, then X is covered by one cycle. This result will be in fact generalised by considering tuples instead of pairs of vertices. Let be the minimum degree in the induced graph <X>. For any , . If , and , then X is covered by at most (p-1) cycles of G. If furthermore , (p-1) cycles are sufficient. So we deduce the following: Let p and t () be two integers. Let G be a 2-connected graph of order n, of minimum degree at least t. If , and , then V is covered by at most cycles, where k is the connectivity of G. If furthermore , (p-1) cycles are sufficient. In particular, if and , then G is hamiltonian. Received April 3, 1998     

Graphs admitting transitive commutative group actions

Let X be a compact metric space, and Homeo(X) be the group consisting of all homeomorphisms from X to X. A subgroup H of Homeo(X) is said to be transitive if there exists a point x∈X such that {k(x):k∈H} is dense in X. In this paper we show that, if X=G is a connected graph, then the following five conditions are equivalent: (1) Homeo(G) has a transitive commutative subgroup; (2) G admits a transitive Z²-action; (3) G admits an edge-transitive commutative group action; (4) G admits an edge-transitive Z²-action; (5) G is a circle, or a k-fold loop with k?2, or a k-fold polygon with k?2, or a k-fold complete bigraph with k?1. As a corollary of this result, we show that a finite connected simple graph whose automorphism group contains an edge-transitive commutative subgroup is either a cycle or a complete bigraph.     

Generalization of matching extensions in graphs (III)

Proposing them as a general framework, Liu and Yu (2001) [6] introduced (n,k,d)-graphs to unify the concepts of deficiency of matchings, n-factor-criticality and k-extendability. Let G be a graph and let n,k and d be non-negative integers such that n+2k+d+2?|V(G)| and |V(G)|−n−d is even. If on deleting any n vertices from G the remaining subgraph H of G contains a k-matching and each k-matching can be extended to a defect-d matching in H, then G is called an (n,k,d)-graph. In this paper, we obtain more properties of (n,k,d)-graphs, in particular the recursive relations of (n,k,d)-graphs for distinct parameters n,k and d. Moreover, we provide a characterization for maximal non-(n,k,d)-graphs.     

Complex eigenvalues of a non-negative matrix with a specified graph

Let A be a non-negative matrix of order n with Perron eigenvalue ? and associated directed graph G. Let m be the length of the longest circuit of G. Theorem: If m=2, all eigenvalues of A are real. If 2<m?n, and if λ=μ+iv is an eigenvalue of A, then $\text{μ+|v|tan(}\text{Π}\text{m}\text{) ? ρ}$.     

Sperner families over a subset

Let |X| = n > 0, |Y| = k > 0, and Y ? X. A family A of subsets of X is a Sperner family of X over Y if A₁A₂ for every pair of distinct members of A and every member of A has a nonempty intersection with Y. The maximum cardinality f(n, k) of such a family is determined in this paper. $\text{f(n,k)=(}\text{n}\text{[n}\text{2]}\text{)?(}\text{?k}\text{[n}\text{2]}\text{)}$.     

Actions of large semigroups and random walks on isometric extensions of boundaries

Let G be a real algebraic semi-simple group, X an isometric extension of the flag space of G by a compact group C. We assume that G is topologically transitive on X. We consider a closed sub-semigroup T of G and a probability measure μ on T such that T is Zariski-dense in G and the support of μ generates T. We show that there is a finite number of T-invariant minimal subsets in X and these minimal subsets are the supports of the extremal μ-stationary measures on X. We describe the structure of these measures, we show the conditional equidistribution on C of the μ-random walk and we calculate the algebraic hull of the corresponding cocycle. A certain subgroup generated by the “spectrum” of T can be calculated and plays an essential role in the proofs.     

LK property for σ-ideals

An ideal J of subsets of a Polish space X has (LK) property whenever for every sequence (An) of analytic sets in X, if lim supn∈HAn∉J for each infinite H then _?n∈G∉J for some infinite G. In this note we present a new class of σ-ideals with (LK) property.     

Sur le changement de base stable des intégrales orbitales pondérées

Let G^′ be a quasi-split connected reductive group over a p-adic field F. Let E be a cyclic extension of F. In the context of cyclic base change, we can attach to G^′ and E a twisted space G^* (in the sense of Labesse). Let G be an inner form of G^*. If G^′ is GL(n), SL(n) or more generally a group which we call L-stable, we define and prove the existence of a non-invariant transfer between the weighted orbital integrals of G and those of G^′. For GL(n), such a transfer has been conjectured by Labesse. The proof is based on previous results of harmonic analysis on Lie algebras and on a generalization of a result of Waldspurger concerning Arthur's (G,M)-families.