ON STAR COLOURING OF M(TM,N),M(TN),M(LN) AND M(SN)

A Star coloring of an undirected graph G is a proper vertex coloring of G in which every path on four vertices contains at least three distinct colors. The Star chromatic number of an undirected graph G, denoted by χs(G) is the smallest integer k for which G admits a star coloring with k colors. In this paper, we obtain the exact value of the Star chromatic number of Middle graph of Tadpole graph, Snake graph, Ladder graph and Sunlet graphs denoted by M[Tm,n], M[Tn], M[Ln] and M[Sn] respectively.


INTRODUCTION AND PRELIMINARIES
Throughout this paper we consider the graph G = (V,E) as a undirected, simple, finite and connected graph with no loops. A vertex coloring (4) of a graph is said to be proper coloring if no two adjacent vertices have the same color. In vertex coloring of G, the set of vertices with same color is known as color class. A proper vertex coloring of a graph is said to be star coloring if the induced subgraph of any two color classes is a collection of stars.
In 1973, Branko Grünbaum (5) introduced the concept of star coloring and also he introduce the notion of star chromatic number. In the beginning, he developed a new concept called acyclic coloring, where it is required that every cycle uses at least 3 colors, so the 2 color induced subgraphs are Forests. Later he established the star coloring concept as a special type of acyclic coloring. His works were developed further by Bondy and Hell (4). A star coloring of a graph is a vertex coloring such that the union of any two color classes does not contain a bicolored path of length 3 and the star chromatic number of a graph is a minimum number of colors which are necessary to star color the graph.
In 2004, Guillaume Fertin et al. (6) developed the exact value of the star chromatic number of different families of graphs such as cycles,trees, 2-dimensional grids, complete bipartite graphs, and outer planar graphs. Further, the authors studied and gave bounds for the star chromatic number of some graphs. Albertson et al. (4) proved that finding the star chromatic number is NPcomplete to find out whether s(G) ≤ 3, even when G is a graph that is both planar and bipartite.
They have also proved that, if G is a graph with minimum degree∆, the s(G) ≤ ∆ ∆ − 1 + 2.
In 1974, The concept of middle graph was introduced by Hamada and Yoshimura (4). The (m,n)-Tadpole graph(2) is a graph is obtained by joining a cycle graph Cm to a path graph Pn with a bridge. It is also known as dragon graph. It is denoted by Tm,n.
A Snake graph (3) is a Eulerian path in the hypercube that has no chords. In other words, any hypercube edge joining snake vertices is a snake edge. It is denoted by Tn.
The Ladder graph (7) is a planar undirected graph with 2n vertices and 3n-2 edges. It is denoted by Ln.
The n-Sunlet graph (1) is the graph on 2n vertices obtained by attaching n pendent edges to the cycle graph Cn. It is denoted by Sn.
From the above coloring procedure, it is clear that no path on 4 vertices is bicolored.

Case (ii): For i ≡ 3(mod 4)
To admit star coloring , for 1 ≤ i ≤ n-3 assign the color sequences 3,4,3,5,3,4,3,5,…3,4,3,5 to the successive vertices of wi and for n-2 ≤ i ≤ n assign the colors 3,4,5 to the respective vertices of wi. It requires five minimum colors to color the vertices of M(Sn) to satisfy the definition of star coloring.

Case (iii): For i ≡ 5(mod 4)
To admit star coloring, for i=5,6 assign the colors 4 and 5 to the vertices of wi respectively, for n-2 ≤ i ≤ n assign the colors 3,4,5 to the respective vertices of wi and for 1 ≤ i ≤ n-3 assign the color sequence 3,4,3,5,3,4,3,5,…3,4,3,5 to the successive vertices of wi. An easy check shows that, it requires five minimum colors to color the vertices of M(Sn) to satisfy the definition of star coloring

Case (iv): For i ≡ 2(mod 4)
To admit star coloring, for i=5,6 assign the colors 4 and 5 to the vertices of wi respectively and for 1 ≤ i ≤ n assign the color sequences 3,4,3,5,3,4,3,5….3,4,3,5 to the successive vertices of wi. It requires five minimum colors to color the vertices of M(Sn) to satisfy the definition of star coloring.
By the above cases, it is clear that, no path on 4 vertices is bicoloured.

CONCLUSION
In this paper, we have found the Star chromatic number for Middle graph of Tadpole graph, Snake graph, Ladder graph and Sunlet graph. Further the results will be extended to obtain the bounds for various graphs.