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 ...

A graph G on a topological space X as its set of vertices is clopen if the edge relation of G is a clopen subset of X2 without the diagonal. We study clopen graphs on Polish spaces in terms of their finite induced subgraphs and obtain information about their cochromatic numbers. In this context we investigate modular profinite graphs, a class of graphs obtained from finite graphs by taking inve...

Graph coloring and its generalizations are useful tools in modeling a wide variety of scheduling and assignment problems. In this paper we review several variants of graph coloring, such as precoloring extension, list coloring, multicoloring, minimum sum coloring, and discuss their applications in scheduling.

We consider two problems, namely Min Split-coloring and Min Cocoloring, that generalize the classical Min Coloring problem by using not only stable sets but also cliques to cover all the vertices of a given graph. We prove the NP-hardness of some cases. We derive approximation results for Min Split-coloring and Min Cocoloring in line graphs, comparability graphs and general graphs. This provide...