A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and every assignment of lists of size four to the vertices of the graph, there exists a linear coloring such that the color of each vertex belongs to the list a...

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...

An adjacent vertex distinguishing edge-coloring of a graph G is a proper edge-coloring of G such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring of G is denoted by χ a (G). We prove that χ a (G) is at most the maximum degree plus 2 if G is a planar graph without isolated edges w...

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...

Distributed Greedy Coloring is an interesting and intuitive variation of the standard Coloring problem. Given an order among the colors, a coloring is said to be greedy if there does not exist a vertex for which its associated color can be replaced by a color of lower position in the fixed order without violating the property that neighbouring vertices must receive different colors. We consider...

An interval total t coloring of a graph G is a total coloring of with colors 1, such that at least one vertex or edge of is colored by color − G 2, , t ... G , 1, 2, , i i t = ... , and the edges incident to each vertex together with are colored by consecutive colors, where is the degree of the vertex in . In this paper interval total colorings of bipartite graphs are investigated. ( ) v V G ∈

It is proved in this note that the Grundy number, T(G), and the ochromatic number, x’(G), are the same for any graph G. An n-coloring of a graph G = (V, E) is a function f from I/ onto N ={1,2,..., n} such that, whenever vertices u and u are adjacent. then f(u) f f(u). An n-coloring is complete if for every pair i,j of integers, 1 5 i 5 j 5 n, there exist a pair U, u of adjacent vertices such t...

In this paper, we study a new coloring parameter of graphs called the gap vertexdistinguishing edge coloring. It consists in an edge-coloring of a graph G which induces a vertex distinguishing labeling of G such that the label of each vertex is given by the difference between the highest and the lowest colors of its adjacent edges. The minimum number of colors required for a gap vertex-distingu...

This paper surveys the most important algorithmic and computational results on the Vertex Coloring Problem (VCP) and its generalizations. The first part of the paper introduces the classical models for the VCP, and discusses how these models can be used and possibly strengthened to derive exact and heuristic algorithms for the problem. Computational results on the best performing algorithms pro...

In a graph G, a vertex is said to dominate itself and all its neighbors. A dominating set of a graph G is a subset of vertices that dominates every vertex of G. The domination number γ(G) is the minimum cardinality of a dominating set of G. A proper coloring of a graph G is a function from the set of vertices of the graph to a set of colors such that any two adjacent vertices have different col...