The λ-backbone coloring is one of the various problems of vertex colorings in graphs. Given an integer λ ≥ 2, a graph G = (V,E), and a spanning subgraph (backbone) H = (V, EH) of G, a λ-backbone coloring of (G,H) is a proper vertex coloring V → {1, 2, ...} of G in which the colors assigned to adjacent vertices in H differ by at least λ. The λ-backbone coloring number BBCλ(G,H) of (G,H) is the s...

A (proper) k-coloring of agraph G is a partition = {V1; V2; : : : ; Vk} of V (G) into k independent sets, called color classes. In a k-coloring , a vertex v∈Vi is called a Grundy vertex if v is adjacent to at least one vertex in color class Vj , for every j, j¡ i. A k-coloring is called a Grundy coloring if every vertex is a Grundy vertex. A k-coloring is called a partial Grundy coloring if eve...

In this paper, we show that for every graph of maximum average degree bounded away from d, any (d + 1)-coloring can be transformed into any other one within a polynomial number of vertex recolorings so that, at each step, the current coloring is proper. In particular, it implies that we can transform any 8-coloring of a planar graph into any other 8-coloring with a polynomial number of recolori...

Background: External beam radiation therapy is often administered to patients implanted with silicone gel breast prosthesis. The aim of this study was to investigate the influence of silicone gel breast prosthesis on photon dose distributions. In the event of recurrence, the oncologist may be forced to irradiate through the prosthetic device. To quantify the dose enhancement or reduction below ...

We describe a graph coloring problem associated with the determination of mathematical derivatives. The coloring instances are obtained as intersection graphs of row partitioned sparse derivative matrices. The size of the graph is dependent on the partition and can be varied between the number of columns and the number of nonzero entries. If solved exactly our proposal will yield a significant ...

We revisit the role of graph coloring in modeling problems that arise in efficient estimation of large sparse Jacobian and Hessian matrices using both finite difference (FD) and automatic differentiation (AD) techniques, in each case via direct methods. For Jacobian estimation using column partitioning, we propose a new coloring formulation based on a bipartite graph representation. This is com...

Abs t r ac t . In an edge-coloring of a graph G = (V, E) each color appears around each vertex at most once. An f-coloring is a generalization of an edge-coloring in which each color appears around each vertex v at most f(v) times where f is a function assigning a natural number f(v) e N to each vertex v E V. In this paper we first give a simple reduction of the f-coloring problem to the ordina...

Motivated by the frequency assignment problem in heterogeneous multihop radio networks, where different radio stations may have different transmission ranges, we introduce two new types of coloring of graphs, which generalize the well-known Distance-k-Coloring. Let G = (V,E) be a graph modeling a radio network, and assume that each vertex v of G has its own transmission radius r(v), a nonnegati...

A set coloring α of a graph G is defined as an assignment of distinct subsets of a finite set X of colors to the vertices of G such that all the colors of the edges which are obtained as the symmetric differences of the sets assigned to their end-vertices are distinct. Additionally, if all the sets on the vertices and edges of G form the set of all nonempty subsets of X, then the coloring α is ...