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Acyclic Edge Colorings of Graphs Benny Sudakov
 

Summary: Acyclic Edge Colorings of Graphs
Noga Alon
Benny Sudakov
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) 16(G) for any graph G (see [2],[10]). We prove that
there exists a constant c such that a (G) (G) + 2 for any graph G whose girth is at least
c(G) log (G), and conjecture that this upper bound for a (G) holds for all graphs G. We also
show that a (G) + 2 for almost all -regular graphs.
1 Introduction
All graphs considered here are finite and simple. A coloring of the vertices of a graph is proper if no
pair of adjacent vertices are colored with the same color. Similarly, an edge-coloring of a graph is
proper if no pair of incident edges are colored with the same color. A proper coloring of the vertices
or edges of a graph G is called acyclic if there is no 2-colored cycle in G. In other words, if the union
of any two color classes induces a subgraph of G which is a forest. The acyclic chromatic number of
G introduced in [7] (see also [8, problem 4.11]), denoted by a(G), is the least number of colors in an
acyclic vertex coloring of G. The acyclic edge chromatic number of G, denoted by a (G), is the least

  

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

 

Collections: Mathematics