It is shown that for any computably enumerable (c.e.) degree w, if w 6= 0, then there is a c.e. degree a such that (a ∨w)′ = a′′ = 0′′ (so a is low2 and a ∨w is high). It follows from this and previous work of P. Cholak, M. Groszek and T. Slaman that the low and low2 c.e. degrees are not elementarily equivalent as partial orderings.

We study sufficient l-degree (1 ≤ l < k) conditions for the appearance of perfect and nearly perfect matchings in k-uniform hypergraphs. In particular, we obtain a minimum vertex degree condition (l = 1) for 3-uniform hypergraphs, which is approximately tight, by showing that every 3-uniform hypergraph on n vertices with minimum vertex degree at least (5/9+o(1)) ` n 2 ́ contains a perfect matchi...

Gallai’s path decomposition conjecture states that the edges of any connected graph on n vertices can be decomposed into at most n+1 2 paths. We confirm that conjecture for all graphs with maximum degree at most five.

We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers D. We also discuss the possible extension to maps of higher genus.

Abstract We provide an analytic expression for the quantity described in the title. Namely, we perform a preferential attachment growth process to generate a scale-free network. At each stage we add a new node with m new links. Let k denote the degree of a node, and N the number of nodes in the network. The degree distribution is assumed to converge to a power-law (for k ≥ m) of the form k and ...

The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the Σk relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k ≥ 7, the Σk relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that Low1 is parameter definable, and we provide a new example of a ∅–...

Having seen a probabilistic verifier for linearity of Boolean functions, we now move on to the goal of general low degree testing. Let us restate what we wish to show: Theorem 1 (Low Degree Testing). There are constants δ, γ, γ′ > 0, such that given the table of a function f : Fm → F and a degree d ≤ δ |F|, there is a probabilistic verifier for the statement “deg f ≤ d”. The verifier is given a...

Spin interactions between two moving Dp-branes are analyzed using the GreenSchwarz formalism of boundary states. This approach turns out to be extremely efficient to compute all the spin effects related by supersymmetry to the leading v4/r7−p term. All these terms are shown to be scale invariant, supporting a matrix model description of supergravity interactions. By employing the LSZ reduction ...

The paper presents several results on edge partitions and vertex partitions of graphs into graphs with bounded size components. We show that every graph of bounded tree-width and bounded maximum degree admits such partitions. We also show that an arbitrary graph of maximum degree four has a vertex partition into two graphs, each of which has components on at most 57 vertices. Some generalizatio...

The harmonic index of a graph G is defined as the sum of weights 2 deg(v)+deg(u) of all edges uv of E(G), where deg(v) denotes the degree of a vertex v in V (G). In this note we generalize results of [L. Zhong, The harmonic index on graphs, Appl. Math. Lett. 25 (2012), 561– 566] and establish some upper and lower bounds on the harmonic index of G.