An acyclic coloring of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of star coloring requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding a...

This article studies a degree-bounded generalization of independent sets called co-k-plexes. Constant factor approximation algorithms are developed for the maximum co-k-plex problem on unit-disk graphs. The related problem of minimum co-k-plex coloring that generalizes classical vertex coloring is also studied in the context of unit-disk graphs. We extend several classical approximation results...

Let Γ be directed strongly connected finite graph of uniform outdegree (constant outdegree of any vertex) and let some coloring of edges of Γ turn the graph into deterministic complete automaton. Let the word s be a word in the alphabet of colors (considered also as letters) on the edges of Γ and let Γs be a mapping of vertices Γ. A coloring is called k-synchronizing if for any word t |Γt| ≥ k ...

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...