The hull and geodetic numbers of orientations of graphs

For an oriented graph D, let I_D[u,v] denote the set of all vertices lying on a u-v geodesic or a v-u geodesic. For S⊆V(D), let I_D[S] denote the union of all I_D[u,v] for all u,v∈S. Let [S]_D denote the smallest convex set containing S. The geodetic number g(D) of an oriented graph D is the minimum cardinality of a set S with I_D[S]=V(D) and the hull number h(D) of an oriented graph D is the minimum cardinality of a set S with [S]_D=V(D). For a connected graph G, let O(G) be the set of all orientations of G, define g⁻(G)=min{g(D):D∈O(G)}, g⁺(G)=max{g(D):D∈O(G)}, h⁻(G)=min{h(D):D∈O(G)}, and h⁺(G)=max{h(D):D∈O(G)}. By the above definitions, h⁻(G)≤g⁻(G) and h⁺(G)≤g⁺(G). In the paper, we prove that g⁻(G)<h⁺(G) for a connected graph G of order at least 3, and for any nonnegative integers a and b, there exists a connected graph G such that g⁻(G)−h⁻(G)=a and g⁺(G)−h⁺(G)=b. These results answer a problem of Farrugia in [A. Farrugia, Orientable convexity, geodetic and hull numbers in graphs, Discrete Appl. Math. 148 (2005) 256-262].     

On the supremum of the pseudocompact group topologies

P is the class of pseudocompact Hausdorff topological groups, and P^′ is the class of groups which admit a topology T such that (G,T)∈P. It is known that every G=(G,T)∈P is totally bounded, so for G∈P^′ the supremum T^∨(G) of all pseudocompact group topologies on G and the supremum T#(G) of all totally bounded group topologies on G satisfy T^∨⊆T#.The authors conjecture for abelian G∈P^′ that T^∨=T#. That equality is established here for abelian G∈P^′ with any of these (overlapping) properties. (a) G is a torsion group; (b) |G|?c²; (c) r₀(G)=|G|=ω|G|; (d) |G| is a strong limit cardinal, and r₀(G)=|G|; (e) some topology T with (G,T)∈P satisfies w(G,T)?c; (f) some pseudocompact group topology on G is metrizable; (g) G admits a compact group topology, and r₀(G)=|G|. Furthermore, the product of finitely many abelian G∈P^′, each with the property T^∨(G)=T#(G), has the same property.     

Sets of periods for automorphisms of compact Riemann surfaces

Let G=〈f〉 be a finite cyclic group of order N that acts by conformal automorphisms on a compact Riemann surface S of genus g≥2. Associated to this is a set A of periods defined to be the subset of proper divisors d of N such that, for some x∈S, x is fixed by f^d but not by any smaller power of f. For an arbitrary subset A of proper divisors of N, there is always an associated action and, if g_A denotes the minimal genus for such an action, an algorithm is obtained here to determine g_A. Furthermore, a set A_max is determined for which g_A is maximal.     

Complete Multipartite Graphs and the Relaxed Coloring Game

Let k be a positive integer, d be a nonnegative integer, and G be a finite graph. Two players, Alice and Bob, play a game on G by coloring the uncolored vertices with colors from a set X of k colors. At all times, the subgraph induced by a color class must have maximum degree at most d. Alice wins the game if all vertices are eventually colored; otherwise, Bob wins. The least k such that Alice has a winning strategy is called the d-relaxed game chromatic number of G, denoted ?? _g ^d (G). It is known that there exist graphs such that ?? _g ⁰(G)?=?3, but ?? _g ¹(G)?>?3. We will show that for all positive integers m, there exists a complete multipartite graph G such that m?????? _g ⁰(G)?<??? _g ¹(G).     

Séparation des orbites coadjointes d'un groupe exponentiel par leur enveloppe convexe

Looking to the separation of irreducible unitary representations of an exponential Lie group G through the image of their moment map, we propose here a new way: instead to extend the moment map to the universal enveloping algebra of G, we define a non linear mapping Φ from the dual of the Lie algebra g of G to the dual g^+∗ of a larger solvable group G⁺, and we extend the representation from G to G⁺, in such a manner that the corresponding coadjoint orbits in g^+∗ have distinct closed convex hull. This allows us to separate the irreducible unitary representations of G.     

Invariant subspaces of some function spaces on a locally compact group

Let G be a σ-compact and locally compact group. If f?L^∞(G) let U_f be the closed subspace of L^∞(G) generated by the left translations of f. Conditions are given which ensure that each function in U_f may be expanded in an essentially unique way as an absolutely convergent series of translations of f. In this case U_f contains subspaces which are isometrically isomorphic to l¹. If G is metrizable and nondiscrete there is a continuum Γ in L^∞(G) such that, for each f?Γ, U_f contains no non-zero continuous function, and for f, g?Γ with f ≠ g, U_f ∩ U_g = {0}. If G is non-compact, metrizable, and non-discrete there is a continuum Γ of bounded continuous functions on G such that, for each f?Γ, U_f contains no non-zero left uniformly continuous function, and for f, g?Γ with f ≠ g, U_f ∩ U_g = {0}. The subspaces U_f above are translation invariant but are not convolution invariant.     

Semigroup crossed products and the induced algebras of lattice-ordered groups

Let (G,G₊) be a quasi-lattice-ordered group with positive cone G₊. Laca and Raeburn have shown that the universal C^∗-algebra C^∗(G,G₊) introduced by Nica is a crossed product BG₊α_×G₊ by a semigroup of endomorphisms. The goal of this paper is to extend some results for totally ordered abelian groups to the case of discrete lattice-ordered abelian groups. In particular given a hereditary subsemigroup H₊ of G₊ we introduce a closed ideal IH₊ of the C^∗-algebra BG₊. We construct an approximate identity for this ideal and show that IH₊ is extendibly α-invariant. It follows that there is an isomorphism between C^∗-crossed products and B₊(G/H)β_×G₊. This leads to our main result that B₊(G/H)β_×G₊ is realized as an induced C^∗-algebra .     

A note on traversing specified vertices in graphs embedded with large representativity

A graph G is said to have property P(2,k) if given any k+2 distinct vertices a,b,v₁,…,v_k, there is a path P in G joining a and b and passing through all of v₁,…,v_k. A graph G is said to have property C(k) if given any k distinct vertices v₁,…,v_k, there is a cycle C in G containing all of v₁,…,v_k. It is shown that if a 4-connected graph G is embedded in an orientable surface Σ (other than the sphere) of Euler genus eg(G,Σ), with sufficiently large representativity (as a function of both eg(G,Σ) and k), then G possesses both properties P(2,k) and C(k).     

A note on domination, girth and minimum degree

Let G be a graph of order n, minimum degree δ?2, girth g?5 and domination number γ. In 1990 Brigham and Dutton [Bounds on the domination number of a graph, Q. J. Math., Oxf. II. Ser. 41 (1990) 269-275] proved that γ?⌈n/2-g/6⌉. This result was recently improved by Volkmann [Upper bounds on the domination number of a graph in terms of diameter and girth, J. Combin. Math. Combin. Comput. 52 (2005) 131-141; An upper bound for the domination number of a graph in terms of order and girth, J. Combin. Math. Combin. Comput. 54 (2005) 195-212] who for i∈{1,2} determined a finite set of graphs G_i such that γ?⌈n/2-g/6-(3i+3)/6⌉ unless G is a cycle or G∈G_i.Our main result is that for every i∈N there is a finite set of graphs G_i such that γ?n/2-g/6-i unless G is a cycle or G∈G_i. Furthermore, we conjecture another improvement of Brigham and Dutton's bound and prove a weakened version of this conjecture.     

Multiplicative infinite loop space theory

Let G be a finitely presented group given by its pre-abelian presentation <X1,…,Xm; Xe11ζ1,…,Xemmζ,ζm+1,…>, where ei≥0 for i = 1,…, m and ζj?G′ for j≥1. Let N be the subgroup of G generated by the normal subgroups [xeii, G] for i = 1,…, m. Then Dn+2(G)≡γn+2(G) (modNG′) for all n≥0, where G” is the second commutator subgroup of G,γn+2(G) is the (n+2)th term of the lower central series of G and Dn+2(G) = G∩(1+△n+2(G)) is the (n+2)th dimension subgroup of G.     

Convex additively slowly varying functions

We study the problem of subtraction of slowly varying functions. It is well-known that the difference of two slowly varying functions need not be slowly varying and we look for some additional conditions which guarantee the slow variation of the difference. To this end we consider all possible decompositions L=F+G of a given increasing convex additively slowly varying function L into a sum of two increasing convex functions F and G. We characterize the class of functions L for which in every such decomposition the summands are necessarily additively slowly varying. The class OΠ₂⁺ we obtain is related to the well-known class OΠ_g where, instead of first order differences as in OΠ_g, we have second order differences.     

Orthogonal factorizations of digraphs

Let G be a digraph with vertex set V(G) and arc set E(G) and let g = (g ⁻, g ⁺) and ƒ = (ƒ ⁻, ƒ ⁺) be pairs of positive integer-valued functions defined on V(G) such that g ⁻(x) ⩽ ƒ ⁻(x) and g ⁺(x) ⩽ ƒ ⁺(x) for each x ∈ V(G). A (g, ƒ)-factor of G is a spanning subdigraph H of G such that g ⁻(x) ⩽ id _H (x) ⩽ ƒ ⁻(x) and g ⁺(x) ⩽ od _H (x) ⩽ ƒ ⁺(x) for each x ∈ V(H); a (g, ƒ)-factorization of G is a partition of E(G) into arc-disjoint (g, ƒ)-factors. Let = {F ₁, F ₂,…, F _m} and H be a factorization and a subdigraph of G, respectively. is called k-orthogonal to H if each F _i , 1 ⩽ i ⩽ m, has exactly k arcs in common with H. In this paper it is proved that every (mg+m−1,mƒ−m+1)-digraph has a (g, f)-factorization k-orthogonal to any given subdigraph with km arcs if k ⩽ min{g ⁻(x), g ⁺(x)} for any x ∈ V(G) and that every (mg, mf)-digraph has a (g, f)-factorization orthogonal to any given directed m-star if 0 ⩽ g(x) ⩽ f(x) for any x ∈ V(G). The results in this paper are in some sense best possible.      

The forcing hull and forcing geodetic numbers of graphs

For every pair of vertices u,v in a graph, a u-v geodesic is a shortest path from u to v. For a graph G, let I_G[u,v] denote the set of all vertices lying on a u-v geodesic. Let S⊆V(G) and I_G[S] denote the union of all I_G[u,v] for all u,v∈S. A subset S⊆V(G) is a convex set of G if I_G[S]=S. A convex hull [S]_G of S is a minimum convex set containing S. A subset S of V(G) is a hull set of G if [S]_G=V(G). The hull number h(G) of a graph G is the minimum cardinality of a hull set in G. A subset S of V(G) is a geodetic set if I_G[S]=V(G). The geodetic number g(G) of a graph G is the minimum cardinality of a geodetic set in G. A subset F⊆V(G) is called a forcing hull (or geodetic) subset of G if there exists a unique minimum hull (or geodetic) set containing F. The cardinality of a minimum forcing hull subset in G is called the forcing hull number f_h(G) of G and the cardinality of a minimum forcing geodetic subset in G is called the forcing geodetic number f_g(G) of G. In the paper, we construct some 2-connected graph G with (f_h(G),f_g(G))=(0,0),(1,0), or (0,1), and prove that, for any nonnegative integers a, b, and c with a+b≥2, there exists a 2-connected graph G with (f_h(G),f_g(G),h(G),g(G))=(a,b,a+b+c,a+2b+c) or (a,2a+b,a+b+c,2a+2b+c). These results confirm a conjecture of Chartrand and Zhang proposed in [G. Chartrand, P. Zhang, The forcing hull number of a graph, J. Combin. Math. Combin. Comput. 36 (2001) 81-94].     

Completing an extension of Tunnell's theorem

We look at a special case of a familiar problem: Given a locally compact group G, a subgroup H and a complex representation π₊ of G how does π₊ decompose on restriction to H. Here G is GL⁺(2,F), where F is a nonarchimedian local field of characteristic not two, K a separable quadratic extension of F, GL⁺(2,F) the subgroup of index 2 in GL(2,F) consisting of those matrices whose determinant is in NK/F(K^∗), π₊ is an irreducible, admissible supercuspidal representation of GL⁺(2,F) and H=K^∗ under an embedding of K^∗ into GL(2,F).     

Iterated conditional expectations

Consider the probability space ([0,1),B,λ), where B is the Borel σ-algebra on [0,1) and λ the Lebesgue measure. Let f=₁[0,1/2) and g=₁[1/2,1). Then for any ε>0 there exists a finite sequence of sub-σ-algebras Gj⊂B(j=1,…,N), such that putting f₀=f and fj=E(fj−1|Gj), j=1,…,N, we have ‖fN−g_‖∞<ε; here E(⋅|Gj) denotes the operator of conditional expectation given σ-algebra Gj. This is a particular case of a surprising result by Cherny and Grigoriev (2007) [1] in which f and g are arbitrary equidistributed bounded random variables on a nonatomic probability space. The proof given in Cherny and Grigoriev (2007) [1] is very complicated. The purpose of this note is to give a straightforward analytic proof of the above mentioned result, motivated by a simple geometric idea, and then show that the general result is implied by its special case.     

The structure of {\phi}-stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature

Suppose (N ⁿ , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as ${P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}Suppose (N ⁿ , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as P(g)=R^m_￥(g)=R(g)-2D_glog f-|?_glog f|_g²{P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}, where dm=f dvol(g){dm=\phi\,dvol(g)} and R(g) is the scalar curvature of (N ⁿ , g). In this paper, under a technical assumption on f{\phi}, we prove that f{\phi}-stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane \mathbbC{\mathbb{C}} or the cylinder \mathbbR×\mathbbS¹{\mathbb{R}\times\mathbb{S}^1}.     

Optimally small sumsets in finite abelian groups

Let G be a finite abelian group of order g. We determine, for all 1?r,s?g, the minimal size μ_G(r,s)=min|A+B| of sumsets A+B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μ_G(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μ_G are recalled in the Introduction.