Pancyclic graphs and a conjecture of Bondy and Chvátal

A p-vertex graph is called pancyclic if it contains cycles of every length l, 3 ≤ l ≤ p. In this paper we prove the following conjecture of Bondy and Chvátal: If a graph G has vertex degree sequence d₁ ≤ d₂ ≤ … ≤ d_ν, and if $\text{d}{}_{\mathrm{k}}\text{≤ k <}\text{p}\text{2}$ implies d_ν?k ≥ p ? k, then G is pancyclic or bipartite.     

The nilpotent π-length of finite π-solvable groups with restrictions on subgroups

V.S. Monakhov’s conjecture concerning an estimate of the nilpotent π-length of a π-solvable group G is confirmed in the case l _p (G) ≤ 1 for all p ∈ π ? {q} and q ∈ π. New estimates of the nilpotent π-length of a π-solvable group with a supersolvable π-Hall subgroup are also obtained.     

Conjugacy classes of non-normal subgroups of finite p-groups

Let G be a finite p-group, and let ν(G) denote the number of conjugacy classes of non-normal subgroups of G. It is known that either ν(G) ≤ 1 or ν(G) ≥ p. We determine all p-groups G with ν(G) ≤ p + 1.     

Hamiltonian cycles in particular k-partite graphs

Let $\text{G}$(itk, p) denote the class of k-partite graphs, where each part is a stable set of cardinality p and where the edges between any pair of stable sets are those of a perfect matching. Maru?i? has conjectured that if G belongs to $\text{G}$(k, p) and is connected then G is hamiltonian. It is proved that the conjecture is true for k ≤ 3 or p ≤ 3; but for k ≥ 4 and p ≥ 4 a non-hamiltonian connected graph in $\text{G}$(k, p) is constructed.     

On strong reality of the unipotent Lie-type subgroups over a field of characteristic 2

A group G is called strongly real if its every nonidentity element is strongly real, i.e. conjugate with its inverse by an involution of G. We address the classical Lie-type groups of rank l, with l ≤ 4 and l ≤ 13, over an arbitrary field, and the exceptional Lie-type groups over a field K with an element η such that the polynomial X ² + X + η is irreducible either in K[X] or K ₀[X] (in particular, if K is a finite field). The following question is answered for the groups under study: What unipotent subgroups of the Lie-type groups over a field of characteristic 2 are strongly real?     

A conjecture on the number of conjugacy classes in ap-solvable group

IfG is ap-solvable group, it is conjectured that k(G/O _P (G) ≤ |G| _p ′. The conjecture is easily obtained for solvable groups as a consequence of R. Knörr’s work on the k(GV) problem. Also, a related result is obtained: k(G/F(G)) is bounded by the index of a nilpotent injector ofG.     

On equality of central and class preserving automorphisms of finite p-groups

Let G be a finite non-abelian p-group, where p is a prime. Let Aut_c(G) and Aut_z(G) respectively denote the group of all class preserving and central automorphisms of G. We give a necessary and sufficient condition for G such that Aut_c(G) = Aut_z(G) and classify all finite non-abelian p-groups G with elementary abelian or cyclic center such that Aut_c(G) = Aut_z(G). We also characterize all finite p-groups G of order ≤ p ⁷ such that Aut_z(G) = Aut_z(G) and complete the classification of all finite p-groups of order ≤ p ⁵ for which there exist non-inner class preserving automorphisms.     

Generators of Simple Lie Algebras and the Lower Rank of Some Pro-P Groups

Abstract

Let  be a simple classical Lie algebra over a field F of characteristic p > 7. We show that > d () = 2, where d() is the number of generators of . Let G be a profinite group. We say that G has lower rank ≤ l, if there are {G _α} open subgroups which from a base for the topology at the identity and each G _α is generated (topologically) by no more than l elements. There is a standard way to associate a Lie algebra L(G) to a finitely generated (filtered) pro-p group G. Suppose L(G) ?  ? tF _p [t], where  is a simple Lie algebra over F _p , the field of p elements. We show that the lower rank of G is ≤ d () + 1. We also show that if  is simple classical of rank r and p > 7 or p 2r ² ? r, then the lower rank is actually 2.     

Variations on a theorem of Petersen

For an (r ? 2)-edge-connected graphG (r ≥ 3) for orderp containing at mostk edge cut sets of cardinalityr ? 2 and for an integerl with 0 ≤l ≤ ?p/2?, it is shown that (1) ifp is even, 0 ≤k ≤ r(l + 1) ? 1, and $$\mathop \sum \limits_{v \in V(G)} |\deg _G v - r|< r(2 + 2l) - 2k$$ , then the edge independence numberβ ₁ (G) is at least (p ? 2l)/2, and (2) ifp is odd, The sharpness of these results is discussed.     

On continuous maps between Grassmann manifolds

LetG _n,k denote the Grassmann manifold ofk-planes in ?ⁿ. We show that for any continuous mapf: G _n,k→G_n,l the induced map inZ/2-cohomology is either zero in positive dimensions or has image in the subring generated by w₁(γ_{n, k}), provided 1≤l<k≤[n/2] andn≥k+2l-1. Our main application is to obtain negative results on the existence of equivariant maps between oriented Grassmann manifolds. We also obtain positive results in many cases on the existence of equivariant maps between oriented Grassmann manifolds.     

An L p -L q analog of miyachi’s theorem for nilpotent lie groups and sharpness problems

The purpose of this paper is to formulate and prove an L ^p -L ^q analog of Miyachi’s theorem for connected nilpotent Lie groups with noncompact center for 2 ≤ p, q ≤ +∞. This allows us to solve the sharpness problem in both Hardy’s and Cowling-Price’s uncertainty principles. When G is of compact center, we show that the aforementioned uncertainty principles fail to hold. Our results extend those of [1], where G is further assumed to be simply connected, p = 2, and q = +∞. When G is more generally exponential solvable, such a principle also holds provided that the center of G is not trivial. Representation theory and a localized Plancherel formula play an important role in the proofs.     

Independent cycles with limited size in a graph

Letk and s be two positive integers with s≥3. LetG be a graph of ordern ≥sk. Writen =qk + r, 0 ≤r ≤k - 1. Suppose thatG has minimum degree at least (s - l)k. Then G containsk independent cyclesC ₁,C ₂,...,C _k such thats ≤l(C _i ) ≤q for 1 ≤i ≤r arnds ≤l(C _i ) ≤q + 1 fork -r <i ≤k, where l(C_i) denotes the length ofC _i .     

The Lindelöf Number of Functional Spaces on Monolithic Compacta

Let X be a compactum, τ be an infinite cardinal, and t(X) ≤ τ. In this case, l(C_p(X)) ≤ 2^τ. If X is τ-monolitliic, then l(C_p(X)) ≤ τ⁺. In addition, if X is zero-dimensional and there are no τ⁺-Aronszajn trees, then l(C_p(X)) ≤ τ.     

Merten's theorem for arithmetic progressions

Let k be an integer ≥ 1 and let l be an integer such that 1 ≤ l ≤ k, (l,k) = 1. An asymptotic formula (valid for large x) is obtained for the product

$\text{∏}\text{p≤x,p≡l(}\text{mod}\text{k)}\text{1?}\text{1}\text{p}$

, generalizing a familiar result of Mertens.     

L p -spaces on locally compact groups

In this paper, some relations between L ^p -spaces on locally compact groups are found. Applying these results proves that for a locally compact group G, the convolution Banach algebras L ^p (G) ∩ L ¹(G) (1 < p ≤ ∞), and A _p (G) ∩ L ¹(G) (1 < p < ∞) are amenable if and only if G is discrete and amenable.     

List‐coloring the square of a subcubic graph

The square G² of a graph G is the graph with the same vertex set G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that every planar graph G with maximum degree Δ(G) = 3 satisfies χ(G²) ≤ 7. Kostochka and Woodall conjectured that for every graph, the list‐chromatic number of G² equals the chromatic number of G², that is, χ_l(G²) = χ(G²) for all G. If true, this conjecture (together with Thomassen's result) implies that every planar graph G with Δ(G) = 3 satisfies χ_l(G²) ≤ 7. We prove that every connected graph (not necessarily planar) with Δ(G) = 3 other than the Petersen graph satisfies χ_l(G²) ≤8 (and this is best possible). In addition, we show that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 7, then χ_l(G²) ≤ 7. Dvo?ák, ?krekovski, and Tancer showed that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 10, then χ_l(G²) ≤6. We improve the girth bound to show that if G is a planar graph with Δ(G) = 3 and g(G) ≥ 9, then χ_l(G²) ≤ 6. All of our proofs can be easily translated into linear‐time coloring algorithms. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 65–87, 2008     

Ghosts in modular representation theory

A ghost over a finite p-group G is a map between modular representations of G which is invisible in Tate cohomology. Motivated by the failure of the generating hypothesis—the statement that ghosts between finite-dimensional G-representations factor through a projective—we define the ghost number of kG to be the smallest integer l such that the composite of any l ghosts between finite-dimensional G-representations factors through a projective. In this paper we study ghosts and the ghost numbers of p-groups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation k, holds for all p-groups. We then compute the ghost numbers of all cyclic p-groups and all abelian 2-groups with C₂ as a summand. We obtain bounds on the ghost numbers for abelian p-groups and for all 2-groups which have a cyclic subgroup of index 2. Using these bounds we determine the finite abelian groups which have ghost number at most 2. Our methods involve techniques from group theory, representation theory, triangulated category theory, and constructions motivated from homotopy theory.     

Labelling of some planar graphs with a condition at distance two

The problem of vertex labeling with a condition at distance two in a graph, is a variation of Hale’s channel assignment problem, which was first explored by Griggs and Yeh. For positive integerp ≥q, the λ _p,q -number of graph G, denoted λ(G;p, q), is the smallest span among all integer labellings ofV(G) such that vertices at distance two receive labels which differ by at leastq and adjacent vertices receive labels which differ by at leastp. Van den Heuvel and McGuinness have proved that λ(G;p, q) ≤ (4q-2) Δ+10p+38q-24 for any planar graphG with maximum degree Δ. In this paper, we studied the upper bound of λ _p ,q-number of some planar graphs. It is proved that λ(G;p, q) ≤ (2q?1)Δ + 2(2p?1) ifG is an outerplanar graph and λ(G;p,q) ≤ (2q?1) Δ + 6p - 4q - 1 if G is a Halin graph.