Acyclic and star coloring problems are specialized vertex coloring problems that arise in the efficient computation of Hessians using automatic differentiation or finite differencing, when both sparsity and symmetry are exploited. We present an algorithmic paradigm for finding heuristic solutions for these two NP-hard problems. The underlying common technique is the exploitation of the structur...

Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. Let C(u) be the set of colors of vertex u and edges incident to u under f . For an IE-total coloring f of G using k colors, if C(u) 6= C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-color...

For integers k, r > 0, a (k, r)-coloring of a graph G is a proper coloring on the vertices of G by k colors such that every vertex v of degree d(v) is adjacent to vertices with at least min{d(v), r} different colors. The dynamic chromatic number, denoted by χ2(G), is the smallest integer k for which a graph G has a (k, 2)-coloring. A list assignment L of G is a function that assigns to every ve...

This paper is concerned with algorithms and complexity results for defective coloring, where a defective (k; d)-coloring is a k coloring of the vertices of a graph such that each vertex is adjacent to at most d-self-colored neighbors. First, (2; d) coloring is shown NP-complete for d 1, even for planar graphs, and (3; 1) coloring is also shown NP-complete for planar graphs (while there exists a...

We investigate a coloring problem, called ordered coloring, in grids and some other families of grid-like graphs. Ordered coloring (also known as vertex ranking) has applications, among other areas, in efficient solving of sparse linear systems of equations and scheduling parallel assembly of products. Our main technical results improve upper and lower bounds for the ordered chromatic number of...

A fall coloring of a graph G is a proper coloring of the vertex set of G such that every vertex of G is a color dominating vertex in G (that is, it has at least one neighbor in each of the other color classes). The fall coloring number χf (G) of G is the minimum size of a fall color partition of G (when it exists). Trivially, for any graph G, χ(G) ≤ χf (G). In this paper, we show the existence ...

We study the problem of transforming one list (vertex) coloring of a graph into another list coloring by changing only one vertex color assignment at a time, while at all times maintaining a list coloring, given a list of allowed colors for each vertex. This problem is known to be PSPACE-complete for bipartite planar graphs. In this paper, we first show that the problem remains PSPACE-complete ...

An oriented graph is 2-stratified if its vertex set is partitioned into two classes, where the vertices in one class are colored red and those in the other class are colored blue. Let H be a 2-stratified oriented graph rooted at some blue vertex. An H-coloring of an oriented graph D is a red-blue coloring of the vertices of D in which every blue vertex v belongs to a copy of H rooted at v in D....

Defective coloring is a variant of traditional vertex-coloring, according to which adjacent vertices are allowed to have the same color, as long as the monochromatic components induced by the corresponding edges have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to th...

Given a graph D = (V (D), A(D)) and a coloring of D, not necessarily a proper coloring of either the arcs or the vertices of D, we consider the complexity of finding a path of D from a given vertex s to another given vertex t with as few different colors as possible, and of finding one with as many different colors as possible. We show that the first problem is polynomial-time solvable, and tha...