### Star-in-coloring of Some Special Graphs

**A. Sugumaran and P. Kasirajan**

Department of Mathematics,
Government Arts College, Tiruvannamalai - 606 603, Tamil Nadu, INDIA.

email: sugumaranaruna@gmail.com,

(Received on: December 18, 2017)

**ABSTRACT**

A proper coloring of a graph G = (V, E) is a mapping f:V → N such that if vivj∈E then f(vi) ± f(vj). In this paper we investigate the lower and upper bounds for the star-in-chromatic number of the graphs such as cycle, regular cyclic, gear, fan, double fan, web and complete binary tree. In addition we have given the general coloring pattern of all these graphs and their star-in-chromatic number.

**AMS Subject Classification:**05C15, 05C20.

**Keywords:**Coloring, Star-in-coloring, Star-in-chromatic number.

**1.INTRODUCTION**

Coloring of graphs is an active research area in graph theory. Many variations of chromatic number of graphs have been studied in literature. The concept of acyclic coloring of graphs was introduced by Grunbaum^{7}, also he introduced the concept of star-coloring of graphs. A graph G is called star-coloring if no path of length three is bicolored. The star- coloring of graphs have been investigated by Fertin, et al.^{5} and Nesetril, et al.^{9}. A digraph G is said to be in-coloring if any path of length two with end vertices of same color are always directed towards the middle vertex. Motivated through the concepts of star-coloring and in- coloring, Sudha and Kanniga^{10} have introduced the star-in-coloring graphs.
For notation and graph theory terminology we refer to Harary^{6}. Let f = (f, f)be a simple, connected digraph with vertex set f = {c1,v_{2}, ..... , vn } and edge set E, each element of E is a directed edge. An orientation of a graph G is obtained by applying an orientation for each edge e = vivj∈ E either from vi to cjor vitovj. We provide the basic definitions of graph theory which are necessary for our present investigations.

**Definition 1.1 **A proper coloring of a graph G is a mapping f: V → {1,2,3, ... } such that if e = vivj∈ E, then f(vi) ± f(vj).

**Definition 1.2 **A graph G is said to be k-colorable if we can assign one of k-colors to each vertex so that adjacent vertices have different colors.
If G is k-colorable, but not (k-1) colorable, then we say that the chromatic number of graph G denoted by X (G) is k.

**Definition 1.3 **A star-coloring of a graph G is a proper coloring of the graph with the condition that no path of length three (P4) is 2-colored.
An n-star-coloring of a graph G is a star-coloring of G using at most n colors.

**Definition 1.4 **An in-coloring of a graph G is a proper coloring of the graph G if there exist any path P_{3} of length two with the end vertices are of the same color, then the edges of P_{3} are oriented towards the central vertex.

**Definition 1.5 **A graph G is said to admit star-in-coloring orientation if it satisfies the following conditions.
1.No path of length three (P4) is bicolored.
2.If any path of length two (P_{3}) with end vertices of same color, then the edges of P_{3} are directed towards the middle vertex.

**Definition 1.6 **The minimum number of colors required for the star-in-coloring of a graph G is called the star-in-chromatic number of G and is denoted by X_{is}(f).

The vertices v_{1} and v_{3} are assigned with the color 1, the vertex v_{4} is assigned with the color 2 and the vertex v_{2} is assigned with the color 3.

In this graph we see that no two adjacent vertices have the same color, no path on four vertices is bicolored, each and every edge in a path of length two in which end vertices have same color are oriented towards the central vertex. Hence it is star-in-colored with orientation. Further the star-in-chromatic number of the above graph is 3.

**Definition 1.7A **graph G = (V, E) in which |V| > 3 and maximum number of chords are drawn without form a triangle and the graph is regular, then the graph is called a regular cyclic graph. We denotea regular cyclic graph with p vertices and degree of each vertex n is denoted by RC(p, n).

**Definition 1.8 **A connected acyclic graph is called a tree. A binary tree is a tree in which only one vertex of degree two and each of the remaining vertices is of degree one or three. A vertex of degree two in a binary tree is called its root vertex. In a binary tree a vertex v is said to be at level l if v is at a distance l from the root vertex.

**Definition 1.9 **A binary tree with level f is said to be complete if each level f of the binary tree contains exactly 2l vertices, where 0 ≤ l ≤ A. A complete binary tree with level n is denoted by BTn.
Note that the complete binary tree BTn contains|V| = 1 + 2 + 2^{2} + ... + 2^{n} vertices and |E| = |V| - 1 edges.

**2. MAIN RESULTS**

In this section, we have to find the lower and upper bounds of star-in-chromatic number of some of the standard graphs. First we find the upper and lower bounds of star-in- chromatic number of cycle C_{n}.

From the above cases, we conclude that 3 ≤ X_{si} ( C_{n} ) ≤ 4.

** Remark: **The star-in-coloring of C_{n} is not possible, when n is odd. Since if we assign alternate colors for consecutive vertices, we obtain the initial and final vertices are of the same colors. On the other hand, if we assign a new color to the final vertex, then the in-coloring property is not satisfied.

** Illustration 2.1.1 **The star-in-coloring of cycle C8 is shown in Figure 2.1.1.

The vertices v_{1},v_{3}, and v_{5} are assigned the color 1,whereas the vertices v_{2}, v_{4} and v_{0} are assigned with colors 2, 3 and 4 respectively.

The star-in-chromatic number of C_{6} is X_{si}(C_{6}) = 4.

** STAR-IN-COLORING OF REGULAR CYCLIC GRAPH**

** Theorem 2.2** The regular cyclic graph RC(p, n) admits star-in-coloring and its star-in- chromaticnumber is n + 1, where p is an even integer and p > 3.

** Proof: **Consider a regular cyclic graph RC(p, n) which consists of 2^{n} vertices and n2 edges,
where n is the degree of the each vertex. Let the vertices be denoted by v_{1},v_{2}, ... , v2n. Define a function f: V → { 1,2, 3, ... } as follows.

The vertices v_{1},v_{3},v_{5} and v_{7} are assigned the color 1. The vertices v_{2} and v_{6} are assigned the color 2. The vertices v_{4} and v8 are assigned the color 3. The vertex v_{0} is assigned the color 4.
The star-in-chromatic number of G_{5} is X_{is}(G_{5})=4.

**Illustration 2.3.2 **Consider a gear graph G4. It consists of 7 vertices and 9 edges.
The vertices v_{1},v_{3} and v_{5} are assigned the color 1. The vertices v_{2},v_{4},v_{6} and v_{0} are assigned the colors 2,3,4 and 5 respectively.

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