An acyclic coloring of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of star coloring requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding a...

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.

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 products of paths and cycles are considered. We determine the acyclic chromatic numbers of three such products: grid graphs...

A graph is called 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we establish a local property of 1-planar graphs which describes the structure in the neighborhood of small vertices (i.e. vertices of degree no more than seven). Meanwhile, some new classes of light graphs in 1-planar graphs with the bounded degree are found. Theref...

An acyclic edge coloring of a graph G is proper such that no bichromatic cycles are produced. The conjecture by Fiamčik (1978) and Alon, Sudakov Zaks (2001) states every simple with maximum degree Δ acyclically ( + 2 ) -colorable. Despite many milestones, the remains open even for planar graphs. In this paper, we confirm affirmatively on graphs without intersecting triangles. We do so first sho...

9 An acyclic coloring is a proper coloring with the additional property that the union of 10 any two color classes induces a forest. We show that every graph with maximum degree at 11 most 5 has an acyclic 7-coloring. We also show that every graph with maximum degree at 12 most r has an acyclic (1 + b (r+1) 2 4 c)-coloring. 13

An acyclic coloring of a graph G is a coloring of its vertices such that:(i) no two neighbors in G are assigned the same color and (ii) no bicolored cycle can exist in G. The acyclic chromatic number of G is the least number of colors necessary to acyclically color G. Recently it has been proved that any graph of maximum degree 5 has an acyclic chromatic number at most 8. In this paper we prese...

We prove that, the acyclic chromatic index a′(G) ≤ 6∆ for all graphs with girth at least 9. We extend the same method to obtain a bound of 4.52∆ with the girth requirement g ≥ 220. We also obtain a relationship between g and a′(G).

An edge colouring of a graph G is called acyclic if it is proper and every cycle contains at least three colours. We show that for every ε > 0, there exists a g = g(ε) such that if G has girth at least g then G admits an acyclic edge colouring with at most (1 + ε)∆ colours.