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A proper edge coloring of a graph is acyclic, if every cycle of the graph has at least 3 colors. Let
*r* be a positive integer. An edge coloring is
*r*-acyclic if it is proper and every cycle
*C* has at least
colors. The
*r*-acyclic edge chromatic
number of a graph
*G* is the minimum number of colors needed for any
*r*-acyclic edge coloring of
*G*. When r=4, the result of this paper is that the 4-acyclic chromatic number of a graph with maximum degree Δ and girth
is less than 18Δ. Furthermore, if the girth of graph
*G* is at least
, then
.

All graphs considered in this paper are finite and simple. A proper edge coloring of a graph

Conjecture 1 (AECC). For every graph G with maximum degree

Given a positive integer r, the r-acyclic edge coloring is a generalization of the acyclic edge coloring of graphs.

An edge coloring is r-acyclic if it is proper and every cycle C has at least

chromatic number

Gerke et al. [

equals to

In this paper we considered the r-acyclic edge coloring problems with r = 4. Using probabilistic arguments, we get some new upper bounds for the 4-acyclic edge chromatic number of arbitrary graph G.

Theorem 1. Let G be a graph with maximum degree

1) If

2) If

We make use of the Lovász Local Lemma as an important tool in our proof. Before giving the proof of Theorem 1, we state the general version of the Lovász Local Lemma (see [

Lemma 2. Let

of the family of events

then

so that with positive probability no event

Proof of Theorem 1.

When

In the first step, we have to prove that there is an edge coloring

1) Every vertex has at most two incident edges of any single color;

2) There are no cycles colored by a single color;

3) There are no cycles colored by just two colors;

4) If the cycle D is colored by just three colors, there are

For each edge

Type I. For each set of three edges

Type II. Given a cycle D of length k, let

Type III. Given a cycle D of length l, let

Type IV. Given a cycle D of length h, let

Obviously, if all the events of Type I, II, III and IV do not occur, then the edge coloring

Let us construct a graph H needed in Lemma 2. Denote X to be a set of three edges or a cycle D in the graph G, where all the three edges are incident with a given vertex and colored by the same color, and all the edges of

{

Lemma 3. Let

1) For each event

2) For each event

3) For each event

4) For each event

Let e be any given edge of graph

Let

of type I, II, III and IV, respectively, where

Let

to be infinity. Define

and

Furthermore, since

ties

In order to prove inequalities (1.1)-(1.4) holds, we just need to show that the following four inequalities (1.5)-

(1.8) hold for every

With the help of the MATLAB calculations, we receive the minimum values of c and corresponding values of

When

when

From the above argument, we know that, there is an edge coloring

Now turn to the second step of our proof. For every color

of G by the edges with the color i. From properties (i) and (ii), we know that

some disjoint paths. Therefore, the edges of

After similar arguments of every color

In a word,

This proof was finished mainly using the Lovász Local Lemma. We believe that with the use of more probabilistic methods, or more careful applications of the Local Lemma, the study of 4-acyclic edge colorings and r- acyclic edge colorings will go further.

YuwenWu,YanXia, (2015) The 4-Acyclic Edge Coloring of Graphs with Large Girths. Journal of Applied Mathematics and Physics,03,1594-1598. doi: 10.4236/jamp.2015.312183