Algorithmic Aspects of Acyclic Edge Colorings A proper coloring of the edges of a graph G is called acyclic if there is no 2-colored cycle Summary: Algorithmic Aspects of Acyclic Edge Colorings Noga Alon Ayal Zaks Abstract A proper coloring of the edges of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a (G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a (G) (G) + 2 where (G) is the maximum degree in G. It is known that a (G) + 2 for almost all -regular graphs, including all -regular graphs whose girth is at least c log . We prove that determining the acyclic edge chromatic number of an arbitrary graph is an NP-complete problem. For graphs G with sufficiently large girth in terms of (G), we present deterministic polynomial time algorithms that color the edges of G acyclically using at most (G) + 2 colors. 1 Introduction All graphs considered here are finite, undirected and simple. A coloring of the edges of a graph is proper if no pair of incident edges are colored with the same color. A proper coloring of the edges of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a (G), is the least number of colors in an acyclic edge coloring of G. The maximum degree in G is denoted by (G). It is known that a (G) 16(G) for any graph G, and that an acyclic edge coloring of G using at most 20(G) can be found efficiently (see [12],[3]). For certain graphs G, a (G) (G) + 2. It Collections: Mathematics