A proper coloring of a graph is a partition of its vertex set into stable sets, where each part corresponds to a color. For a vertex-weighted graph, the weight of a color is the maximum weight of its vertices. The weight of a coloring is the sum of the weights of its colors. Guan and Zhu defined the weighted chromatic number of a vertex-weighted graph G as the smallest weight of a proper colori...

For a given induced hereditary property P, a P-coloring of a graph G is an assignment of one color to each vertex such that the subgraphs induced by each of the color classes have property P. We consider the effectiveness of on-line P-coloring algorithms and give the generalizations and extensions of selected results known for on-line proper coloring algorithms. We prove a linear lower bound fo...

A b-coloring of a graph G by k colors is a proper k-coloring of the vertices 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 φ(G) of a graph G is the maximum k for which G has a b-coloring by k colors. This concept was introduced by R.W. Irving and D.F. Manlove in 1999. In this paper we study the b-chromatic nu...

An acyclic coloring of a graph is a proper vertex coloring without bichromatic cycles. We show that the acyclic colorings of any weakly chordal graph G correspond to the proper colorings of triangulations of G. As a consequence, we obtain polynomial-time algorithms for the acyclic coloring problem and the perfect phylogeny problem on the class of weakly chordal graphs. Our results also imply li...

An explicit coloring of the edges of Kn is constructed such that every copy of K4 has at least four colors on its edges. As n →∞, the number of colors used is n. This improves upon the previous bound of O(n) due to Erdős and Gyárfás obtained by probabilistic methods. The exponent 1/2 is optimal, since it is known that at least Ω(n) colors are required in such a coloring. The coloring is related...

Given a coloring of the vertices, we say subgraph H is monochromatic if every vertex of H is assigned the same color, and rainbow if no pair of vertices of H are assigned the same color. Given a graph G and a forbidden graph F , we define an F -WORM coloring of G as a coloring of the vertices if G without a rainbow or monochromatic subgraph isomorphic to F . We explore such colorings especially...

Let G be a connected simple graph. A restrained dominating set S of the vertex set of G, V (G) is a secure restrained dominating set of G if for each u ∈ V (G) \ S, there exists v ∈ S such that uv ∈ E(G) and the set (S \ {v}) ∪ {u} is a restrained dominating set of G. The minimum cardinality of a secure restrained dominating set of G, denoted by γsr(G), is called the secure restrained dominatio...

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number b(G) of a graph G is the largest integer k such that G admits a b-coloring with k colors. A simple graph G is called b-vertex (edge) critical if the removal of any vertex (edge) of G increases its b-chromatic number. In thi...

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes. El-Sahili and Kouider have conjectured that every d-regular graph with girth at least 5 has a b-coloring with d+1 colors. We show that the Petersen graph infirms this conjecture, and we propose a new formulation of this question and give a positive a...

In the paper we consider distributed algorithms for greedy graph coloring. For the largest-first (LF) approach, we propose a new distributed algorithm which is shown to color a graph in an expected time of O(∆ log n log∆) rounds, and we prove that any distributed LF-coloring algorithm requires at least Ω(∆) rounds. We discuss the quality of obtained colorings in the general case and for particu...