An acyclic k-coloring of a graph G is a proper vertex coloring of G which uses at most k colors such that the graph induced by the union of every two color classes is a forest. In this paper, we mainly prove that every 5-connected graph with maximum degree five is acyclically 8-colorable, improving partially [5].

An acyclic k-coloring of a graph G is a proper vertex coloring of G, which uses at most k colors, such that the graph induced by the union of every two color classes is a forest. In this note, we prove that every graph with maximum degree six is acyclically 11-colorable, thus improving the main result of [12].

In this paper, we introduce the new notion of acyclic improper colorings of graphs. An improper coloring of a graph G is a mapping c from the set of vertices of G to a set of colors such that for every color i, the subgraph induced by the vertices with color i satisses some property depending on i. Such an improper coloring is acyclic if for every two distinct colors i and j, the subgraph induc...

The oriented chromatic number χo(~ G) of an oriented graph ~ G = (V,A) is the minimum number of vertices in an oriented graph ~ H for which there exists a homomorphism of ~ G to ~ H. The oriented chromatic number χo(G) of an undirected graph G is the maximum of the oriented chromatic numbers of all the orientations of G. This paper discusses the relation between the oriented chromatic number an...

A graph is 1-planar if it can be drawn on the plane in such a way that every edge crosses at most one other edge. We prove that the acyclic chromatic number of every 1-planar graph is at most 20.

We study edge coloring games defining the so-called game chromatic index of a graph. It has been reported that the game chromatic index of trees with maximum degree ∆ = 3 is at most ∆ + 1. We show that the same holds true in case ∆ ≥ 6, which would leave only the cases ∆ = 4 and ∆ = 5 open.

An edge-coloring of a graph G with natural numbers is called a sum edge-coloring if the colors of edges incident to any vertex of G are distinct and the sum of the colors of the edges of G is minimum. The edge-chromatic sum of a graph G is the sum of the colors of edges in a sum edge-coloring of G. It is known that the problem of finding the edge-chromatic sum of an r-regular (r ≥ 3) graph is N...