In the minimum sum coloring problem, the goal is to color the vertices of a graph with the positive integers such that the sum of all colors is minimal. Recently, it was shown that coloring a graph by iteratively coloring maximum independent sets yields a 4 + o(1) approximation for the minimum sum coloring problem. In this note, we show that this bound is tight. We construct a graph for which t...

A vertex-distinguishing coloring of a graph G consists in an edge or a vertex coloring (not necessarily proper) of G leading to a labeling of the vertices of G, where all the vertices are distinguished by their labels. There are several possible rules for both the coloring and the labeling. For instance, in a set irregular edge coloring [5], the label of a vertex is the union of the colors of i...

We propose the notion of a majority $k$-edge-coloring graph $G$, which is an edge-coloring $G$ with $k$ colors such that, for every vertex $u$ at most half edges incident have same color. show best possible results that minimum degree least $2$ has $4$-edge-coloring, and $4$ $3$-edge-coloring. Furthermore, we discuss natural variation edge-colorings some related open problems.

Given a graphG, a total k-coloring ofG is a simultaneous coloring of the vertices and edges ofGwith at most k colors. If ∆(G) is themaximum degree ofG, then no graph has a total ∆-coloring, but Vizing conjectured that every graph has a total (∆ + 2)-coloring. This Total Coloring Conjecture remains open even for planar graphs. This article proves one of the two remaining planar cases, showing th...

Dvořák and Postle [8] introduced aDP-coloring of a simple graph as a generalization of a list-coloring. They proved a Brooks’ type theorem for a DP-coloring, and Bernsheteyn, Kostochka and Pron [5] extended it to a DP-coloring of multigraphs. However, detailed structure when a multigraph does not admit a DP-coloring was not specified in [5]. In this note, we make this point clear and give the c...

This paper presents selective coloring as a new paradigm for branch-and-bound exact maximum clique search. Approximate coloring has, in recent, years been at the heart of leading solvers in the field. Selective coloring proposes to relax coloring up to a certain threshold. The result is a less informed but lighter decision heuristic. Different operators for the remaining uncolored vertices give...

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) is related to conflict-free coloring and other traditional coloring problems. Such coloring problems can model (among others) efficient frequency assignments in cellular networks. Our main technical results improve upper and lower boun...

In this paper we propose a deterministic parallel graph coloring algorithm that enables Multi-Coloring in parallel for sparse undirected graphs by coarse-grained segmentation and auxiliary graph coloring. The proposed algorithm is implemented and tested on standard problem instances from engineering applications and benchmarked against various relevant deterministic graph coloring algorithms. Q...

Graph vertex coloring is one of the most studied NP-hard combinatorial optimization problems. Given the hardness of the problem, various heuristic algorithms have been proposed for practical graph coloring, based on local search, population-based approaches and hybrid methods. The research in graph coloring heuristics is very active and improved results have been obtained recently, notably for ...