For an integer k ≥ 1, a graph G is k-colorable if there exists a mapping c : VG → {1, . . . , k} such that c(u) 6= c(v) whenever u and v are two adjacent vertices. For a fixed integer k ≥ 1, the k-COLORING problem is that of testing whether a given graph is k-colorable. The girth of a graph G is the length of a shortest cycle in G. For any fixed g ≥ 4 we determine a lower bound `(g), such that ...

Let h = ∑ hαβX Y β be a polynomial with complex coefficients. The Lojasiewicz exponent of the gradient of h at infinity is the upper bound of the set of all real λ such that |gradh(x, y)| ≥ c|(x, y)| in a neighbourhood of infinity in C, for c > 0. We estimate this quantity in terms of the Newton diagram of h. The equality is obtained in the nondegenerate case.

We estimate the expected number of limit cycles situated in a neighbourhood of the origin of a planar polynomial vector field. Our main tool is a distributional inequality for the number of zeros of some families of univariate holomorphic functions depending analytically on a parameter. We obtain this inequality by methods of Pluripotential Theory. This inequality also implies versions of a str...

A new methodology for a robust analysis of uncertain nonlinear dynamic systems is presented in this paper. The originality of the method proposed lies in the combination of the centre manifold theory with the polynomial chaos approach. The first one is known to be a powerful tool for model reduction of nonlinear dynamic systems in the neighbourhood of the Hopf bifurcation point while the polyno...

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An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the vertex set of the graph. The minimum weight efficient domination problem is the problem of finding an efficient dominating set of minimum weight in a given vertex-weighted graph; the maximum weight efficient domination problem is defined similarly. We develop a framework...

We describe a new algorithm to compute the geometric intersection number between two curves, given as edge vectors on an ideal triangulation. Most importantly, this algorithm runs in polynomial time in the bit-size of the two edge vectors. In its simplest instances, this algorithm works by finding the minimal position of the two curves. We achieve this by phrasing the problem as a collection of...

In this paper, we introduce the new notion of acyclic improper colorings of graphs. An improper coloring of a graph is a vertex-coloring in which adjacent vertices are allowed to have the same color, but each color class V i satisses some condition depending on i. Such a coloring is acyclic if there are no alternating 2-colored cycles. We prove that every outerplanar graph can be acyclically 2-...

DESIGNING DISTANCE-PRESERVING FAULT-TOLERANT TOPOLOGIES By Sitarama Swamy Kocherlakota The objective of designing a fault-tolerant communication network (CN) is to maintain its functionality in the presence of failures. The approaches taken in designing fault-tolerant systems can be classified, in general, by the functionality criteria. There are two functionality criteria that have received mu...