Int. J. Math. And Appl., 6(1)(2018), 203–212. (Special Issue) ISSN: 2347-1557 Available Online: http://ijmaa.in/ A p p l i c a t i o n s • I S S N : 2 3 4 7 - 1 5 5 7 • I n t e r n a t i o n a l J o u r n a l o f M a t h e m a t i c s A n d i t s International Journal of Mathematics And its Applications Star Colouring of Line Graph Formed From the Cartesian Product of cycle and Path graphs L. Jethruth Emelda Mary 1, * and T. Monica 1 1 PG & Research Department of Mathematics, St.Joseph’s College of Arts and Science, Cuddalore, Tamil Nadu, India. Abstract: In this paper we determined the star chromatic number of the line graph which is formed from the Cartesian product of path and cycle graphs. Also the relationship between the chromatic number and star chromatic number are analysed and illustrated with some examples. MSC: 05C90. Keywords: Path graph, Cycle graph, Cartesian product of simple graph, Line graph, Proper colouring, Chromatic number, Star- colouring, Star chromatic number. c  JS Publication. Accepted on: 13 th April 2018 1. Introduction In this paper, we have taken the graph to be undirected, finite and simple graph.The concept of Star Colouring was introduced by Grunbaum in 1973. In 2003, Nesetril and Ossona de Mendez proved that the star chromatic number to be bounded on each proper minor closed class [2]. The complexity of star colouring was exhibited by Albertson et al in 2004. Cartesian product of graphs have been described by Whitehead and Russel in 1912 according to Imrich and Klavzar [5]. The concept of line graph was invented by H. Whitney in 1932. Here we constructed the Line graph L(G) formed from the Cartesian product of cycle and path graphs and we obtained the bounds for L(G) using star colouring for various non-negative values of m and n. We begin with some basic definitions and notations. Definition 1.1. The Cartesian product of simple graphs G and H is denoted by G × H whose vertex set is V (G) × V (H) and u =(u1,u2) and v =(v1,v2) are adjacent if u1 = v1 and u2 is adjacent to v2 in H or u1 is adjacent to v1 in G and u2 = v2 in H. Example 1.2. * E-mail: jethruth15@ gmail.com