A graph is called equistable when there is a non-negative weight function on its vertices such that a set S of vertices has total weight 1 if and only if S is maximal stable. We characterize those series-parallel graphs that are equistable, generalizing results of Mahadev, Peled and Sun about equistable outer-planar graphs.

We prove that the number of maximal points in a random sample taken uniformly and independently from a convex polygon is asymptotically normal in the sense of convergence in distribution. Many new results for other planar regions are also derived. In particular, precise Poisson approximation results are given for the number of maxima in regions bounded above by a nondecreasing curve.

We show that the maximal number of singular moves required to pass between any two regularly homotopic planar or spherical curves with at most n crossings, grows quadratically with respect to n. Furthermore, this can be done with all curves along the way having at most n + 2 crossings.

We prove that every n-vertex graph of genus 9 and maximal degree k has an edge separator of size O( ..;gTffl). The upper bound is best possible to within a constant factor. This extends known results on planar graphs and similar results about vertex separators. We apply the edge separatorto the isoperimetric number problem, graph embeddings and lower bounds for crossing numbers.

Some topics of our twenty some years of joint work is discussed. Just to name a few; joint behavior of the maximum of the Wiener process and its location, global and local almost sure limit theorems, strong approximation of the planar local time difference, a general Strassen type theorem, maximal local time on subsets. AMS 2000 Subject Classification: Primary 60J15, Secondary 60F15, 60J55.

In this paper, we present some results on maximal planar graphs with minimum degree ve, denoted by MPG5 graphs [6]. We consider a subset of MPG5 graphs, called the Z graphs, for which all vertices of degree superior to ve are not adjacent. We give a vertex four coloring for every Z graph.

In the title compound, C(12)H(15)NO, the C=C and C=O functional groups and the benzene ring are involved in an extended conjugated system. The mol-ecules are essentially planar with a maximal deviation from planarity for the non-H atoms of 0.062 (2) Å.

A bstract We derive analytically the terms of maximal transcendentality planar 2- and 3-point functions single-trace chiral primary operators $$ \mathcal{N} N = 2 SQCD on ℝ 4 , to all orders in ’t Hooft coupling. These results prove two conjectures we formulated previous work. Furthermore, also provide an exp...

Single sodium channels from human brain cortex tissue were incorporated into voltage-clamped planar lipid bilayers in the presence of batrachotoxin and studied with various doses of the new anaesthetic compound propofol (2,6-diisopropylphenol). Propofol was found to depress two major sodium channel functions, leading to a reduction of the time-averaged fractional channel open-time (half-maximal...

We consider vertex colorings of graphs in which each color has an associated cost which is incurred each time the color is assigned to a vertex. The cost of the coloring is the sum of the costs incurred at each vertex. The cost chromatic number of a graph with respect to a cost set is the minimum number of colors necessary to produce a minimum cost coloring of the graph. We show that the cost c...