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 G is a proper coloring of the vertex set of G such that G contains no bichromatic cycles. The acyclic chromatic number of a graph G is the minimum number k such that G has an acyclic coloring with k colors. In this paper, acyclic colorings of Hamming graphs, products of complete graphs, are considered.

For integers k, r > 0, a (k, r)-coloring of a graph G is a proper coloring on the vertices of G by k colors such that every vertex v of degree d(v) is adjacent to vertices with at least min{d(v), r} different colors. The dynamic chromatic number, denoted by χ2(G), is the smallest integer k for which a graph G has a (k, 2)-coloring. A list assignment L of G is a function that assigns to every ve...

A well-known special case of a conjecture attributed to Ryser (actually appeared in the thesis of Henderson [7]) states that k-partite intersecting hypergraphs have transversals of at most k−1 vertices. An equivalent form of the conjecture in terms of coloring of complete graphs is formulated in [1]: if the edges of a complete graph K are colored with k colors then the vertex set of K can be co...

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 ...

Let f be a nonnegative integer valued function on the vertex set of graph. A graph is strictly f-degenerate if each nonempty subgraph ? has v such that deg?(v)<f(v). In this paper, we define new concept, transversal, which generalizes list coloring, signed DP-coloring, L-forested-coloring, and (f1,f2,…,fs)-partition. cover G H with V(H)=?v?V(G)Xv, where Xv={(v,1),(v,2),…,(v,s)}; edge M=?uv?E(G)...

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 ...

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 χb(G) of a graph G is the largest integer k such that G admits a b-coloring by k colors. We present some lower bounds for the b-chromatic number of connected bipartite graphs. We also d...

For integers r and k > 0 (k > r), a conditional (k, r)-coloring of a graph G is a proper k-coloring of G such that every vertex v of G has at least min{r, d(v)} differently colored neighbors, where d(v) is the degree of v. Given an r, for a graph G we are interested in obtaining a conditional coloring using minimum k — this value is denoted by χ r (G), called the r th order conditional chromati...

Let G be a simple, undirected graph. We say that two edges of G are within distance 2 of each other if either they are adjacent or there is some other edge that is adjacent to both of them. A distance-2-edge-coloring of G is an assignment of colors to edges so that any two edges within distance 2 of each other have distinct colors, or equivalently, a vertex-coloring of the square of the line gr...