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