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 ...

We consider the sum coloring (chromatic sum) and sum multi-coloring problems for restricted families of graphs. In particular, we consider the graph classes of proper intersection graphs of axis-parallel rectangles, proper interval graphs, and unit disk graphs. All the above mentioned graph classes belong to a more general graph class of (k+1)clawfree graphs (respectively, for k = 4, 2, 5). We ...

In a previous work, we proposed a new integer programming formulation for the graph coloring problem which, to a certain extent, avoids symmetry. We studied the facet structure of the 0/1-polytope associated with it. Based on these theoretical results, we present now a Branch-and-Cut algorithm for the graph coloring problem. Our computational experiences compare favorably with the well-known ex...