Total domination of graphs and small transversals of hypergraphs

The main result of this paper is that every 4-uniform hypergraph on n vertices and m edges has a transversal with no more than (5n + 4m)/21 vertices. In the particular case n = m, the transversal has at most 3n/7 vertices, and this bound is sharp in the complement of the Fano plane. Chvátal and McDiarmid [5] proved that every 3-uniform hypergraph with n vertices and edges has a transversal of size n/2. Two direct corollaries of these results are that every graph with minimal degree at least 3 has total domination number at most n/2 and every graph with minimal degree at least 4 has total domination number at most 3n/7. These two bounds are sharp.     

On the inner curvature of the second fundamental form of helicoidal surfaces

Let M be a helicoidal surface in E ³, free of points of vanishing Gaussian curvature. Let H be the mean curvature and K _II the curvature of the second fundamental form. In this note it is shown that the helicoidal surfaces satisfying K _II =H are locally characterized by constancy of the ratio of the principal curvatures. Moreover it is proved that these helicoidal surfaces are determined by a first order differential equation. Research supported by E.E.C. contract CHRX-CT92-0050.     

A note on a spanning 3-tree

A tree T is called a k-tree, if the maximum degree of T is at most k. In this paper, we prove that if G is an n-connected graph with independence number at most n + m + 1 (n≥1,n≥m≥0), then G has a spanning 3-tree T with at most m vertices of degree 3.     

The normal density of prime ideals in small regions

For a Gaussian prime (i) define ()=min|–| where runs through Gaussian primes satisfying ||>||. We prove that, subject to the Riemann Hypothesis for appropriateL-functions

Independent systems of representatives in weighted graphs

The following conjecture may have never been explicitly stated, but seems to have been floating around: if the vertex set of a graph with maximal degree Δ is partitioned into sets V _i of size 2Δ, then there exists a coloring of the graph by 2Δ colors, where each color class meets each V _i at precisely one vertex. We shall name it the strong 2Δ-colorability conjecture. We prove a fractional version of this conjecture. For this purpose, we prove a weighted generalization of a theorem of Haxell, on independent systems of representatives (ISR’s). En route, we give a survey of some recent developments in the theory of ISR’s. The research of the first author was supported by grant no 780/04 from the Israel Science Foundation, and grants from the M. & M. L. Bank Mathematics Research Fund and the fund for the promotion of research at the Technion. The research of the third author was supported by the Sacta-Rashi Foundation.     

Convexity andq-gamma function

Inequalities and convexity properties known for the gamma function are extended to theq-gamma function, 0<q<1. Applying some classical inequalities for convex functions, we deduce monotonicity results for several functions involving theq-gamma function. Further applications to the probability theory are given.     

Klingenberg chain spaces

We propose a definition ofKlingenberg chain space, motivated by examples that are obtained from the action of the linear group on the projective line over an algebra over a local ring.     

Dynamic Multi-level Overlay Graphs for Shortest Paths

Multi-level overlay graphs represent a speed-up technique for shortest paths computation which is based on a hierarchical decomposition of a weighted directed graph G. They have been shown to be experimentally efficient, especially when applied to timetable information. However, no theoretical result on the cost of constructing, maintaining and querying multi-level overlay graphs in a dynamic environment is known. In this paper, we show theoretical properties of multi-level overlay graphs that lead us to the definition of a new data structure for the computation and the maintenance of an overlay graph of G while weight decrease or weight increase operations are performed on G. Our solution is theoretically faster than the recomputation from scratch and allows queries that can be performed more efficiently than running Dijkstra’s shortest paths algorithm on G. This work was partially supported by the Future and Emerging Technologies Unit of EC (IST priority – 6th FP), under contract no. FP6-021235-2 (project ARRIVAL).     

Note on limits of finite graphs

We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.     

When does the giant component bring unsatisfiability?

We study random constraint satisfaction problems using the wide class of models introduced by the author [36], which includes various forms of random SAT and other well-studied problems. We determine precisely which of these models remain almost surely satisfiable when the number of clauses is increased beyond the point at which a giant component appears in the underlying constraint hypergraph. This work is supported by an NSERC Research Grant and a Sloan Research Fellowship. Much of this work was done while the author was a Visiting Researcher at Microsoft Research.     

On a stability theorem of liapunov

A complex function theory argument enables us to give a simple and elegant proof of a stability theorem of Liapunov.     

Connected but not path-connected subspaces of infinite graphs

Solving a problem of Diestel [9] relevant to the theory of cycle spaces of infinite graphs, we show that the Freudenthal compactification of a locally finite graph can have connected subsets that are not path-connected. However we prove that connectedness and path-connectedness to coincide for all but a few sets, which have a complicated structure.     

Lipschitz regularity of the minimizers of autonomous integral functionals with discontinuous non-convex integrands of slow growth

Let be a Borelian function and let (P) be the problem of minimizing

Note making a <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-free graph bipartite

We show that every K ₄-free graph G with n vertices can be made bipartite by deleting at most n ²/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdős. Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred P. Sloan fellowship.     

Birth control for giants

The standard Erdős-Rényi model of random graphs begins with n isolated vertices, and at each round a random edge is added. Parametrizing n/2 rounds as one time unit, a phase transition occurs at time t = 1 when a giant component (one of size constant times n) first appears. Under the influence of statistical mechanics, the investigation of related phase transitions has become an important topic in random graph theory. We define a broad class of graph evolutions in which at each round one chooses one of two random edges {v ₁, v ₂}, {v ₃, v ₄} to add to the graph. The selection is made by examining the sizes of the components of the four vertices. We consider the susceptibility S(t) at time t, being the expected component size of a uniformly chosen vertex. The expected change in S(t) is found which produces in the limit a differential equation for S(t). There is a critical time t _c so that S(t) → ∞ as t approaches t _c from below. We show that the discrete random process asymptotically follows the differential equation for all subcritical t < t _c . Employing classic results of Cramér on branching processes we show that the component sizes of the graph in the subcritical regime have an exponential tail. In particular, the largest component is only logarithmic in size. In the supercritical regime t > t _c we show the existence of a giant component, so that t = t _c may be fairly considered a phase transition. Computer aided solutions to the possible differential equations for susceptibility allow us to establish lower and upper bounds on the extent to which we can either delay or accelerate the birth of the giant component. Research supported by the Australian Research Council, the Canada Research Chairs Program and NSERC. Research partly carried out while the author was at the Department of Mathematics and Statistics, University of Melbourne.     

Mutually Independent Hamiltonian Connectivity of (n, k)-Star Graphs

A graph G is hamiltonian connected if there exists a hamiltonian path joining any two distinct nodes of G. Two hamiltonian paths and of G from u to v are independent if u = u ₁ = v ₁, v = u _v(G) = v _v(G) , and u _i ≠ v _i for every 1 < i < v(G). A set of hamiltonian paths, {P ₁, P ₂, . . . , P _k }, of G from u to v are mutually independent if any two different hamiltonian paths are independent from u to v. A graph is k mutually independent hamiltonian connected if for any two distinct nodes u and v, there are k mutually independent hamiltonian paths from u to v. The mutually independent hamiltonian connectivity of a graph G, IHP(G), is the maximum integer k such that G is k mutually independent hamiltonian connected. Let n and k be any two distinct positive integers with n–k ≥ 2. We use S _n,k to denote the (n, k)-star graph. In this paper, we prove that IHP(S _n,k ) = n–2 except for S _4,2 such that IHP(S _4,2) = 1.      

Some properties of almostS-continuous functions

Noiri et al. in [16] introduced and investigated the notion of almosts-continuous functions. The object of this paper is to investigate some more properties of this type of functions.     

Proof of the bandwidth conjecture of Bollobás and Komlós

In this paper we prove the following conjecture by Bollobás and Komlós: For every γ > 0 and integers r ≥ 1 and Δ, there exists β > 0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least ((r ? 1)/r + γ)n and H is an r-chromatic graph with n vertices, bandwidth at most β n and maximum degree at most Δ, then G contains a copy of H.     

The superstar packing problem

Hell and Kirkpatrick proved that in an undirected graph, a maximum size packing by a set of non-singleton stars can be found in polynomial time if this star-set is of the form {S ₁, S ₂, ..., S _k } for some k∈ℤ₊ (S _i is the star with i leaves), and it is NP-hard otherwise. This may raise the question whether it is possible to enlarge a set of stars not of the form {S ₁, S ₂, ..., S _k } by other non-star graphs to get a polynomially solvable graph packing problem. This paper shows such families of depth 2 trees. We show two approaches to this problem, a polynomial alternating forest algorithm, which implies a Berge-Tutte type min-max theorem, and a reduction to the degree constrained subgraph problem of Lovász. Research is supported by OTKA grants K60802, TS049788 and by European MCRTN Adonet, Contract Grant No. 504438.