A k-edge-coloring of a graph G = (V, E) is a function c that assigns an integer c(e) (called color) in {0, 1, · · · , k−1} to every edge e ∈ E so that adjacent edges get different colors. A k-edge-coloring is linear compact if the colors incident to every vertex are consecutive. The problem k − LCCP is to determine whether a given graph admits a linear compact k-edge coloring. A k-edge-coloring...

Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) bean ordered partition of V (G) into color classes. For any vertex v of G, define the color code c_Π(v) of v with respect to Π to be a k -tuple (d(v, R_1), d(v, R_2), . . . , d(v, R_k)), where d(v, R_i) is the min{d(v, x)|x ∈ R_i}. If distinct vertices have distinct color codes, then we call f a locat...

We present new algorithms for k-coloring and minimum (x(G)-) coloring random and semi-random kcolorable graphs in polynomial expected time. The random graphs are drawn from the G(n,p, k) model and the semi-random graphs are drawn from the G s ~ ( n , p , k) model. In both models, an adversary initially splits the n vertices into IC color classes, each of size @(n). Then the edges between vertic...

In this paper we investigate a variation of the graph coloring game, as studied in [2]. In the original coloring game, two players, Alice and Bob, alternate coloring vertices on a graph with legal colors from a fixed color set, where a color α is legal for a vertex if said vertex has no neighbors colored α. Other variations of the game change this definition of a legal color. For a fixed color ...

The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The List k-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u) ⊆ {1, . . . , k}. Let Pn denote the path on n vertices, and G + H and rH the disjoint union of two graphs G and H and r copies of H, r...

For positive integers k and r , a (k, r)-coloring of a graphG is a proper coloring of the vertices with k colors such that every vertex of degree i will be adjacent to vertices with at least min{i, r} different colors. The r-dynamic chromatic number of G, denoted by χr (G), is the smallest integer k for which G has a (k, r)-coloring. For a k-list assignment L to vertices of G, an (L, r)-colorin...

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of Grundy Coloring, the problem of determining whether a given graph has Grundy number at least k. We also study the variants Weak Grundy Coloring (where the coloring is not necessarily proper) and Connected Grundy Col...

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...

Graph coloring has been studied for a long time and continues to receive interest within the research community [43]. It has applications in scheduling [46], timetables, and compiler register allocation [45]. The most popular variant of graph coloring, k-coloring, can be thought of as an assignment of k colors to the vertices of a graph such that adjacent vertices are assigned different colors....