Point partition numbers and generalized Nordhaus-Gaddum problems
Let be a simple graph. For a general edge coloring of a graph (i.e., not necessarily a proper edge coloring) and a vertex of , denote by the set (not a multiset) of colors used to color the edges incident to . For a general edge coloring of a graph , if for any two different vertices and of , then we say that is a point-distinguishing general edge coloring of . The minimum number of colors required for a point-distinguishing general edge coloring of , denoted by , is called...
A subset of the vertex set of a graph is called point-set dominating, if for each subset there exists a vertex such that the subgraph of induced by is connected. The maximum number of classes of a partition of , all of whose classes are point-set dominating sets, is the point-set domatic number of . Its basic properties are studied in the paper.
The asymptotic distributions of the number of vertices of a given degree in random graphs, where the probabilities of edges may not be the same, are given. Using the method of Poisson convergence, distributions in a general and particular cases (complete, almost regular and bipartite graphs) are obtained.
Starting with the computation of the appropriate Poisson kernels, we review, complement, and compare results on drifted Laplace operators in two different contexts: homogeneous trees and the hyperbolic half-plane.
Suppose that red and blue points occur as independent homogeneous Poisson processes in ℝd. We investigate translation-invariant schemes for perfectly matching the red points to the blue points. For any such scheme in dimensions d=1, 2, the matching distance X from a typical point to its partner must have infinite d/2th moment, while in dimensions d≥3 there exist schemes where X has finite exponential moments. The Gale–Shapley stable marriage is one natural matching scheme, obtained by iteratively...
On définit localement la notion de polyèdre de rang deux pour un polyèdre fini de dimension deux à courbure négative ou nulle. On montre que le revêtement universel d’un tel espace est soit le produit de deux arbres, soit un immeuble de Tits euclidien de rang deux.
This article considers the eta power . It is proved that the coefficients of in this expression, as polynomials in b, exhibit equidistribution of the coefficients in the nonzero residue classes mod 5 when n = 5j+4. Other symmetries, as well as symmetries for other primes and prime powers, are proved, and some open questions are raised.