The pre-coloring extension problem consists, given a graph G and a subset of nodes to which some colors are already assigned, in nding a coloring of G with the minimum number of colors which respects the pre-coloring assignment. This can be reduced to the usual coloring problem on a certain contracted graph. We prove that pre-coloring extension is polynomial for complements of Meyniel graphs. W...

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 symmetry breaking. The proposed algorithm is implemented and tested on standard problem instances from engineering applications and benchmarked against various deterministic graph coloring algorithms. Quantified result...

We study interval coloring problems and present new upper and lower bounds for several variants. We are interested in four problems, online coloring of intervals with and without bandwidth and a new problem called lazy online coloring again with and without bandwidth. We consider both general interval graphs and unit interval graphs. Specifically, we establish the difference between the two mai...

We define a variant of the H-coloring problem where the number of preimages of certain vertices is predetermined as part of the problem input. We consider the decision and the counting version of the problem, namely the restrictive H-coloring and the restrictive #H-coloring problems, and we provide a dichotomy theorem determining the H’s for which the restrictive H-coloring problem is either NP...

Let G(V,E) be a k-uniform hypergraph. A hyperedge e ∈ E is said to be properly (r, p) colored by an r-coloring of vertices in V if e contains vertices of at least p distinct colors in the rcoloring. An r-coloring of vertices in V is called a strong (r, p) coloring if every hyperedge e∈E is properly (r, p) colored by the r-coloring. We study the maximum number of hyperedges that can be properly ...

All graph in this paper be a connected and simple graph. Let c:V(G)→{1,2,…,k} is proper vertex coloring where k ≥ 2 which induces edge c':E(G)→{1,2,…,k} define by c' (uv)=|c(u)-c(v)|, uv E(G) called graceful k-coloring. A c of G if k-coloring for some k∈ N. The minimum chromatic number denoted χ_g (G). In paper, we will investigate the establish exact value on grid family namely H 〖(H〗_n) n≥2 m...

In this paper we present a hybrid technique that applies a genetic algorithm followed by wisdom of artificial crowds approach to solving the graph-coloring problem. The genetic algorithm described here utilizes more than one parent selection and mutation methods depending on the state of fitness of its best solution. This results in shifting the solution to the global optimum more quickly than ...