A Grundy k-coloring of a graph G, is a vertex k-coloring of G such that for each two colors i and j with i < j, every vertex of G colored by j has a neighbor with color i. The Grundy chromatic number Γ(G), is the largest integer k for which there exists a Grundy k-coloring for G. In this note we first give an interpretation of Γ(G) in terms of the total graph of G, when G is the complement of a...

Let G be a graph and R ⊆ V (G). A proper edge-coloring of a graph G with colors 1, . . . , t is called an R-sequential t-coloring if the edges incident to each vertex v ∈ R are colored by the colors 1, . . . , dG(v), where dG(v) is the degree of the vertex v in G. In this note, we show that if G is a graph with ∆(G) − δ(G) ≤ 1 and χ′(G) = ∆(G) = r (r ≥ 3), then G has an R-sequential r-coloring ...

An adjacent vertex distinguishing edge-coloring of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors χa(G) required to give G an adjacent vertex distinguishing coloring is studied for graphs with no isolated edge. We prove χa(G) ≤ 5 for such graphs with maximum degree Δ(G) = 3 and prove χa(G) ≤ Δ(G) ...

In a properly vertex-colored graphG, a path P is a rainbow path if no two vertices of P have the same color, except possibly the two end-vertices of P . If every two vertices of G are connected by a rainbow path, then G is vertex rainbow-connected. A proper vertex coloring of a connected graph G that results in a vertex rainbow-connected graph is a vertex rainbow coloring ofG. The minimum numbe...

0957-4174/$ see front matter 2011 Elsevier Ltd. A doi:10.1016/j.eswa.2011.01.098 ⇑ Corresponding author. E-mail addresses: [email protected] (J. Akbari ac.ir (M.R. Meybodi). Vertex coloring problem is a combinatorial optimization problem in which a color is assigned to each vertex of the graph such that no two adjacent vertices have the same color. Cellular learning automata (CLA) is an e...

Given a graph G = (V,E) and positive integral vertex weights w : V → N, the max-coloring problem seeks to find a proper vertex coloring of G whose color classes C1, C2, . . . , Ck, minimize ∑k i=1 maxv∈Ciw(v). The problem arises in scheduling conflicting jobs in batches and in minimizing buffer size in dedicated memory managers. In this paper we present three approximation algorithms and one in...

A basic randomized coloring procedure has been used in probabilistic proofs to obtain remarkably strong results on graph coloring. These results include the asymptotic version of the List Coloring Conjecture due to Kahn, the extensions of Brooks’ Theorem to sparse graphs due to Kim and Johansson, and Luby’s fast parallel and distributed algorithms for graph coloring. The most challenging aspect...

Graph coloring is used to characterize some properties of graphs. A b-coloring of a graph G (using colors 1,2,...,k) is a coloring of the vertices of G such that (i) two neighbors have different colors (proper coloring) and (ii) for each color class there exists a dominating vertex which is adjacent to all other k-1 color classes. In this paper, we build on b-coloring of a graph to propose a ne...

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...

A Grundy coloring of a graph G is a proper vertex coloring of G where any vertex x, colored with c(x), has a neighbor of any color 1, 2, . . . , c(x)− 1. A central graph Gc is obtained from G by adding an edge between any two non adjacent vertices in G and subdividing any edge of G once. In this note we focus on Grundy colorings of central graphs. We present some bounds related to parameters of...