Abs t r ac t . In an edge-coloring of a graph G = (V, E) each color appears around each vertex at most once. An f-coloring is a generalization of an edge-coloring in which each color appears around each vertex v at most f(v) times where f is a function assigning a natural number f(v) e N to each vertex v E V. In this paper we first give a simple reduction of the f-coloring problem to the ordina...

9 An acyclic coloring is a proper coloring with the additional property that the union of 10 any two color classes induces a forest. We show that every graph with maximum degree at 11 most 5 has an acyclic 7-coloring. We also show that every graph with maximum degree at 12 most r has an acyclic (1 + b (r+1) 2 4 c)-coloring. 13

In this paper, we study the coloring problem for the undirected binary de Bruijn interconnection network. The coloring scheme is simple and fast. We propose the coloring algorithm by using the pseudo shortestpath spanning tree rooted at (0 00). Each processor can find its color number by its own identity. Then, based on our coloring algorithm, we propose a 1-fair alternator. Our design is optim...

A set coloring α of a graph G is defined as an assignment of distinct subsets of a finite set X of colors to the vertices of G such that all the colors of the edges which are obtained as the symmetric differences of the sets assigned to their end-vertices are distinct. Additionally, if all the sets on the vertices and edges of G form the set of all nonempty subsets of X, then the coloring α is ...

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...

A polychromatic k-coloring of a plane graph G is an assignment of k colors to the vertices of G such that every face of G has all k colors on its boundary. For a given plane graph G, we seek the maximum number k such that G admits a polychromatic k-coloring. We call a k-coloring in the classical sense (i.e., no monochromatic edges) that is also a polychromatic k-coloring a strong polychromatic ...

A total coloring of a graph G is to color all vertices and edges of G so that no two adjacent or incident elements receive the same color. Let C be a set of colors, and let ω be a cost function which assigns to each color c in C a real number ω(c) as a cost of c. A total coloring f of G is called an optimal total coloring if the sum of costs ω( f (x)) of colors f (x) assigned to all vertices an...