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 ...

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $n$. A $k$-vertex weightings of a graph $G$ is a mapping $w: V(G) to {1, ldots, k}$. A $k$-vertex weighting induces an edge labeling $f_w: E(G) to N$ such that $f_w(uv)=w(u)+w(v)$. Such a labeling is called an {it edge-coloring k-vertex weightings} if $f_{w}(e)not= f_{w}(echr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39'...

If a graph G contains no subgraph isomorphic to some graph H, then G is called H-free. A coloring of a graph G = (V,E) is a mapping c : V → {1, 2, . . .} such that no two adjacent vertices have the same color, i.e., c(u) 6= c(v) if uv ∈ E; if |c(V )| ≤ k then c is a k-coloring. The Coloring problem is to test whether a graph has a coloring with at most k colors for some integer k. The Precolori...

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...

We consider the question of computing the strong edge coloring, square graph coloring, and their generalization to coloring the k power of graphs. These problems have long been studied in discrete mathematics, and their “chaotic” behavior makes them interesting from an approximation algorithm perspective: For k = 1, it is well-known that vertex coloring is “hard” and edge coloring is “easy” in ...

Let G = (V,E) be a graph. A k-coloring for G is a function f : V → [k] such that f(u) 6= f(v) for all (u, v) ∈ E. In other words, a k-coloring is an assignment of vertices to k colors such that no edge is monochromatic. We say that a graph G is k-colorable if there exists a k-coloring for G. The chromatic number of G is the least k such that G is k-colorable. Given a k-colorable graph G, findin...

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 ...

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 b-coloring of a graph G by k colors is a proper k-coloring of G such that in each color class there exists a vertex having neighbors in all the other k− 1 color classes. The b-chromatic number of a graph G, denoted by φ(G), is the maximum k for which G has a b-coloring by k colors. It is obvious that χ(G) ≤ φ(G). A graph G is b-continuous if for every k between χ(G) and φ(G) there is a b-colo...

Given a nite directed graph, a coloring of its edges turns the graph into a nite-state automaton. A k-synchronizing word of a deterministic automaton is a word in the alphabet of colors at its edges that maps the state set of the automaton at least on k-element subset. A coloring of edges of a directed strongly connected nite graph of a uniform outdegree (constant outdegree of any vertex) is k-...