Further Results on Acyclic Chromatic Number *

An acyclic coloring of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees.The purpose of this paper is to derive exact values of acyclic chromatic number of some graphs.


Introduction
Graph coloring is a branch of graph theory which deals with such partitioning problems.For example, suppose that we have world map and we would like to color the countries so that if two countries share a boundary line, then they need to get different colors.We can translate the map to graph by letting countries be represented by vertices and two vertices are made adjacent if and only if the corresponding countries share a boundary line.Then the problem of map coloring is equivalent to vertex coloring of the corresponding graph.Hence the original map coloring now reduces to vertex coloring of the associated graph.
Coloring of a graph is an assignment of colors to the elements like vertices or edges or faces (regions) of a graph.It is said to be a proper coloring, if no two adjacent elements are assigned the same color.The most common types of graph colorings are vertex coloring, edge coloring and face coloring.
A vertex coloring of a graph is an assignment of colors to its vertices so that no two vertices have the same color.The chromatic number of a graph is the minimum number of colors needed to label the vertices, so that adjacent vertices receive different colors.
A proper vertex coloring of a graph is acyclic if every cycle uses at least three colors [1].The acyclic chromatic number of denoted by , G  , a G is the minimum colors required for its acyclic coloring.

Central Graph [2]
Let be a finite undirected graph with no loops and .p C G p q    For any   , p q graph there exists exactly p vertices of degree 1 p  and q vertices of degree 2 in  .
C G  The central graph of two isomorphic graphs is also isomorphic.

Theorem
The acyclic coloring of central graph of cycle, Let be the central graph of which is obtained by sub dividing each edge of exactly once and joining non adjacent vertices of Let the newly introduced vertices be Now assign a proper coloring to the vertices as follows.The coloring is in such a way that the sub graph induced by any two color is a forest containing at most the path The vertices are as-4 .


The newly 2,3 4,5 are assigned the colors and respectively and all others are colored properly.all others are assigned so that the coloring is proper.Now the coloring is obviously acyclic, by the very arrangement of the colors.It is also minimum, because if we replace any color by an already used color, it will become either improper or cyclic (Figures 1 and 2).

Acyclic Coloring of Line Graph of Central
Graph of K

 
L G are adjacent whenever the corresponding lines of are.G

Structural Properties of Line Graph of Central Graph of n K
Line graph of central graph of  There is a cycle C of length 2n with alternate edges from each of the complete graph 1 .
n K 

Theorem
For any complete graph , , , , , where 1, 2,3, , ; Here there exist a unique bridge between each pair of sub graphs 1 .
The bridge in the consecutive pairs of sub graph  is given by for . In a similar manner bridges are formed in non consecutive pairs also.Consider the color class , , , , .
 Assign the color i to the vertex Next we prove that the coloring is acyclic.That is the coloring does not induce a bi-chromatic cycle.Clearly for each complete sub graph 1 j n K  the coloring is acyclic (it never induce a bi-chromatic cycle).Now exactly two pairs of sub graphs 1 , never allow to induce a bi-chromatic cycle for any pair as there is only a unique bridge between each pair of sub graphs 1 .
j n K  Note that bichromatic cycle is possible only for even cycles.The coloring is in such a way that more than three sub graphs 1 .

j n K  never allow to induce a bi-chromatic cycle for any pair
The maximum number of times a color will occur in any bi-chromatic path in this coloring is three.So the above said coloring acyclic.Also the coloring is minimum, as M G in case one of following holds: 1) , x y are in and

Theorem
The acyclic chromatic number of the middle graph of is , , , Then in the middle graph, there are -vertices of degree and another -vertices of degree 4. c The coloring is minimum, as for any cycle minimum 3 olors needed for its acyclic coloring.The coloring is acyclic (Figure 4).

 
V G and x is adjacent to in 2) is in and y  E G, , x y are incident in .G

Some Structural Properties of Total Graph of n C
 Every cycle has a 4 -regular total graph. The number of vertices in the total graph of n C is 2 times the number of vertices in the cycle .

Theorem
The acyclic chromatic number of the total graph of is , , , , , , multiple edges.The central graph of a graph G , G   C G is obtained by subdividing each edge of exactly once and joining all the non-adjacent vertices of simple graph, then by the definition of   C G of a graph. The number of vertices in the central graph of G is    

Figure 2 .
Figure 2. Acyclic coloring of central graph of C 5 .
Let n K be the complete graph on vertices.Consider its line graph of central graph n

K
 , minimum 1 n  colors are required for its proper coloring (Figure3).

Figure 3
Figure 3. Acyclic coloring of

Figure 4 .
Figure 4. Acyclic coloring of middle graph of .C 8 The number of edges in the total graph of n C is 4 times the number of edges in the cycle .n C  The total graph of n C is Eulerian. The total graph of n C is Hamiltonian.