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 ...

If G and H are two cubic graphs, then an H-coloring of G is a proper edge-coloring f with edges of H , such that for each vertex x of G, there is a vertex y of H with f(∂G(x)) = ∂H(y). If G admits an H-coloring, then we will write H ≺ G. The Petersen coloring conjecture of Jaeger states that for any bridgeless cubic graph G, one has: P ≺ G. The second author has recently introduced the Sylveste...

Batch scheduling of conflicting jobs is modeled by batch coloring of a graph. Given an undirected graph and the number of colors required by each vertex, we need to find a proper batch coloring of the graph, namely, to partition the vertices to batches which are independent sets and assign to each batch a contiguous set of colors, whose size is equal to the maximum color requirement of any vert...

A parity walk in an edge-coloring of a graph is a walk along which each color is used an even number of times. Let p(G) be the least number of colors in an edge-coloring of G having no parity path (a parity edge-coloring). Let p̂(G) be the least number of colors in an edge-coloring of G having no open parity walk (a strong parity edge-coloring). Always p̂(G) ≥ p(G) ≥ χ′(G). We prove that p̂(Kn) = ...

A total coloring of a graph G is to color all vertices and edges of G so that no two adjacent or incident elements receive the same color. Let C be a set of colors, and let ω be a cost function which assigns to each color c in C a real number ω(c) as a cost of c. A total coloring f of G is called an optimal total coloring if the sum of costs ω( f (x)) of colors f (x) assigned to all vertices an...

Graph coloring is the de facto standard technique for register allocation within a compiler. In this paper we examine the importance of the quality of the coloring algorithm and various extensions of the basic graph coloring technique by replacing the coloring phase of the GNU compiler’s register allocator with an optimal coloring algorithm. We then extend this optimal algorithm to incorporate ...