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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

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics