L(d,2,1)-Labeling of Simple Graphs
An L(3, 2, 1)-labeling is a simplified model for the channel assignment problem. It is a natural generalization of the widely studied L(2, 1)labeling. An L(3, 2, 1)-labeling of a graph G is a function f from the vertex set V (G) to the set of positive integers such that for any two vertices x, y, if d(x, y) = 1, then |f(x) − f(y)| ≥ 3; if d(x, y) = 2, then |f(x) − f(y)| ≥ 2; and if d(x, y) = 3, then |f(x) − f(y)| ≥ 1. The L(3, 2, 1)-labeling number k(G) of G is the smallest positive integer k such that G has an L(3, 2, 1)-labeling with k as the maximum label. In this paper we determine the L(3, 2, 1)-labeling number for paths, cycles, caterpillars, nary trees, complete graphs and complete bipartite graphs. We also present an upper bound for k(G) in terms of the maximum degree of G.
Optimal L(δ1, δ2, 1)-labeling of eight-regular grids
Given a graph G=(V,E), an L(δ1,δ2,δ3)-labeling is a function f assigning to nodes of V colors from a set {0,1,…,kf} such that |f(u)−f(v)|⩾δi if u and v are at distance i in G. The aim of the L(δ1,δ2,δ3)-labeling problem consists in finding a coloring function f such that the value of kf is minimum. This minimum value is called λδ1,δ2,δ3(G). 
 
In this paper we study this problem on the eight-regular grids for the special values (δ1,δ2,δ3)=(3,2,1) and (δ1,δ2,δ3)=(2,1,1), providing optimal labelings. Furthermore, exploiting the lower bound technique, we improve the known lower bound on λ3,2,1 for triangular grids.
Rapid Spatially Coupled AVO Inversion
The Bayesian methodology yields a natural framework for many geophysical inverse problem, see e.g. (Tarantola and Valette, 1982; Tarantola, 1987; Duijndam, 1988a; Duijndam, 1988b). The solution of a Bayesian inverse problem is represented by the posterior distribution. From the posterior distribution the best estimate and the corresponding uncertainty can be extracted and a set of plausible solutions can be drawn.