The k-core of a graph is the largest subgraph of minimum degree at least k. We show that for k sufficiently large, the (k + 2)-core of a random graph G(n, p) asymptotically almost surely has a spanning k-regular subgraph. Thus the threshold for the appearance of a k-regular subgraph of a random graph is at most the threshold for the (k + 2)-core. In particular, this pins down the point of appea...

The asymptotic properties of random regular graphs are objects of extensive study in mathematics and physics. In this paper we argue, using the theory of spin glasses in physics, that in random regular graphs the maximum cut size asymptotically equals the number of edges in the graph minus the minimum bisection size. Maximum cut and minimal bisection are two famous NP-complete problems with no ...

The notion of a bead metric space is defined as a nice generalization of the uniformly convex normed space such as $CAT(0)$ space, where the curvature is bounded from above by zero. In fact, the bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces and normed bead spaces are identical with uniformly convex spaces. In this paper, w...

Let G be a K4-free 4-regular connected graph with & vertices. Then G contains an independent set of vertices of cardinality at least (7&&4) 26. Furthermore, the proof will yield a polynomial-time algorithm which will return an independent set of cardinality at least (7&&4) 26 for any such graph. A family of K4 -free 4-regular connected graphs with maximum independent set of cardinality 3 11& is...

In this paper, we study the large-scale structure of dense regular graphs. This involves the notion of robust expansion, a recent concept which has already been used successfully to settle several longstanding problems. Roughly speaking, a graph is robustly expanding if it still expands after the deletion of a small fraction of its vertices and edges. Our main result allows us to harness the us...

A model for cleaning a graph with brushes was recently introduced. We consider the minimum number of brushes needed to clean d-regular graphs in this model, focusing on the asymptotic number for random d-regular graphs. We use a degree-greedy algorithm to clean a random d-regular graph on n vertices (with dn even) and analyze it using the differential equations method to find the (asymptotic) n...

The vacuum Einstein equations in five dimensions are shown to admit a solution describing a stationary asymptotically flat spacetime regular on and outside an event horizon of topology S1xS2. It describes a rotating "black ring." This is the first example of a stationary asymptotically flat vacuum solution with an event horizon of nonspherical topology. The existence of this solution implies th...

We study the equations of black strings in spacetimes of arbitrary dimensions with a negative cosmological constant and construct numerically non uniform black strings solutions. Our results suggest the existence of a localised black hole in asymptotically locally AdS spacetime. We also present evidences for a dependence of the critical dimension on the horizon radius.The critical dimension rep...

We present a procedure that transforms singular, asymptotically flat, static and spherically symmetric black hole into regular spacetime. The is solution of General Relativity theory coupled to nonlinear electrodynamics (NLED), even if the original metric not vacuum, or electro-vacuum, (GR).

Numerical arguments are presented for the existence of spherically symmetric regular and black hole solutions of the EYMH equations with a negative cosmological constant. These solutions approach asymptotically the anti-de Sitter spacetime. The main properties of the solutions and the differences with respect to the asymptotically flat case are discussed. The instability of the gravitating spha...