Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 252 https://internationalpubls.com Evaluating Star Vertex Cochromatic Number in Prism, Sunlet and Derived Graphs R. Sabitha1*, V. Kowsalya2 1Research Scholar, 2Associate Professor, PG & Research Department of Mathematics, Sri Ramakrishna College of Arts & Science (Autonomous), Formerly SNR Sons College, Coimbatore. E-Mail ID: 1*sabitha.r@srcas.ac.in, 2kowsalya@srcas.ac.in Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: In this discussion, the star cochromatic number π‘ž is found for the following graphs: Prism Graph π‘ž[π‘Œπ‘š], Line Graph of Prism Graph π‘ž[𝐿(π‘Œπ‘š)], Middle Graph of Prism Graph π‘ž[𝑀(π‘Œπ‘š)], Sunlet Graph π‘ž[π‘†π‘š], Line Graph of Sunlet Graph π‘ž[𝐿(π‘†π‘š)], Middle Graph of Sunlet Graph π‘ž[𝑀(π‘†π‘š)]. Keywords: Prism Graph, Sunlet Graph, Line Graph, Middle Graph, Star Cocoloring, Star Cochromatic Index. AMS Classification: 05C07, 05C15, 05C69, 05C76. 1. Introduction This study examines finite, simple and undirected graphs. Whitney initially presented the concept of a line graph in 1932 [1]. In 1983, Maria Chudnovsky [5] proposed using line graphs as a very fundamental lesson in graph theory. T. Hamada and I. Yoshimura were the ones who first suggested the center graph of the graph [6]. Vertex partitioning is an important concept in graph theory. Partitioning and imposing on vertex sets yielded new concepts and outcomes. In 1977, Lesinak and H. Straight introduced the concept Cocoloring and found some basic results [3]. Kowsalya. V, Vernold Vivin. J and Venkatachalam. M [2] found the star chromatic number for sunlet graph families and their line, middle, central and total graphs. Vernold Vivin. J, Kowsalya. V, and Vimal Kumar. S [7] found the star chromatic number for prism graph families and their derived graphs. Star Cocoloring concept was introduced by M. Poobalaranjani[4]. A clique is defined as a subset W of a simple graph G = (V, E) that produces a complete subgraph of G. If W has cardinality K, it is referred to as k-clique. A subset U of V is considered independent if it creates an empty subgraph of G. If U has cardinality K, it is referred to k-independent set. 2. Preliminaries 2.1: Prism Graph: The Prism graph is a planar, polyhedral graph that resembles the skeleton of an m- prism. An m-prism graph is the same as the generalised Petersen graph 𝑃{𝑛,1} with 2m vertices and 3m edges. It is denoted by π‘Œπ‘š 2.2: Sunlet Graph: The sunlet graph on 2m vertices is obtained by attaching n pendant edges to the cycle 𝐢𝑛 and it is denoted by π‘†π‘š. mailto:1*sabitha.r@srcas.ac.in mailto:2kowsalya@srcas.ac.in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 253 https://internationalpubls.com 2.3: Line Graph: Line graph L(G) is β€’ L(G) is a graph in which each vertex corresponds to an edge of G. β€’ Two L(G) vertices are said to be adjacent if their respective lines share an endpoint in G. 2.4: Middle Graph: The newly added middle vertices of G's surrounding edges are joined to form the Middle Graph M(G). It is made by accurately splitting each edge of G once. 2.5: Star Cocoloring: Let G be a graph k,l,r be non-negative integers then a (k,l,r) - s - cocoloring of G is a partition of the vertex set of G into sets 𝐼1, 𝐼2, 𝐼3, … , πΌπ‘˜, 𝐢1, 𝐢2, 𝐢3, … , 𝐢𝑙 , 𝑆1, 𝑆2, 𝑆3, … , π‘†π‘Ÿ such that πΌπ‘˜ is an independent set each 𝐢𝑙 is a clique and each π‘†π‘Ÿ is a star 𝐾{1,𝑑} where 𝑑 β‰₯ 3. 2.6: Star Cochromatic Number: The Star Cochromatic Number π‘ž is defined as π‘§βˆ—(𝐺) = min{π‘ž: π‘‘β„Žπ‘’π‘Ÿπ‘’ 𝑒π‘₯𝑖𝑠𝑑𝑠 π‘˜, 𝑙, π‘Ÿ π‘ π‘’π‘β„Ž π‘‘β„Žπ‘Žπ‘‘ π‘˜, 𝑙 β‰₯ 0 , π‘Ÿ β‰₯ 1; 𝐺 𝑖𝑠 (π‘˜, 𝑙, π‘Ÿ) βˆ’ 𝑠 βˆ’ π‘π‘œπ‘™π‘œπ‘Ÿπ‘Žπ‘π‘™π‘’ π‘Žπ‘›π‘‘ π‘˜ + 𝑙 + π‘Ÿ = π‘ž } 3. Main Results: 3.1: Star Cochromatic Number of Prism Graphs Theorem 3.1.1: The Star Cochromatic Number of Prism Graph π‘ž[π‘Œπ‘š] for π‘š β‰₯ 3 is π‘ž[π‘Œπ‘š] = ⌈ π‘š 2 βŒ‰ Proof: Let π‘Œπ‘š be prism graph with 2m vertices and 3m edges and 𝑉[π‘Œπ‘š] = {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š}. Consider the color class 𝐢 = {𝑐1, 𝑐2, 𝑐3, … , 𝑐 ⌈ π‘š 2 βŒ‰ }. Define mapping 𝜎: {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š} β†’ π‘π‘˜ βˆ€ π‘˜ = 1,2,3, … Case-(i): π’Ž ≑ 𝟎 π’Žπ’π’… πŸ’ β€’ 𝜎(𝑒4π‘˜, 𝑒4π‘˜βˆ’1, 𝑒4π‘˜+1, 𝑣4π‘˜) = 𝑐2π‘˜ β€’ 𝜎(𝑣4π‘˜βˆ’2, 𝑣4π‘˜βˆ’3, 𝑣4π‘˜βˆ’1, 𝑒4π‘˜βˆ’2) = 𝑐2π‘˜βˆ’1 Case-(ii): π’Ž ≑ 𝟏 π’Žπ’π’… πŸ’ β€’ 𝜎(𝑒4π‘˜βˆ’2, 𝑒4π‘˜βˆ’1, 𝑒4π‘˜βˆ’3, 𝑣4π‘˜βˆ’2) = 𝑐2π‘˜βˆ’1 β€’ 𝜎(𝑣4π‘˜, 𝑣4π‘˜+1, 𝑣4π‘˜βˆ’1, 𝑒4π‘˜) = 𝑐2π‘˜ β€’ 𝜎(𝑣1, π‘’π‘š) = 𝑐 ⌈ π‘š 2 βŒ‰ Case-(iii): π’Ž ≑ 𝟐 π’Žπ’π’… πŸ’ Assign the coloring as follows: β€’ 𝜎(𝑒4π‘˜+1, 𝑒4π‘˜, 𝑒4π‘˜+2, 𝑣4π‘˜+1) = 𝑐2π‘˜ β€’ 𝜎(𝑣4π‘˜+3, 𝑣4π‘˜+2, 𝑣4π‘˜+4, 𝑒4π‘˜+3) = 𝑐2π‘˜βˆ’1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 254 https://internationalpubls.com There exists an uncolored set of vertices {𝑣1, 𝑣2, 𝑣3, 𝑣4, π‘£π‘š , 𝑒1, 𝑒2, 𝑒3}. In which the vertices {𝑣2, 𝑣1, 𝑣3, 𝑒2} forms π‘˜1,3 which is to be colored with 𝑐 ⌈ π‘š 2 βŒ‰βˆ’1 and the vertices {𝑒1, 𝑒3, 𝑣4, π‘’π‘š} forms an independent set is colored with new color 𝑐 ⌈ π‘š 2 βŒ‰ Case-(iv): π’Ž ≑ πŸ‘ π’Žπ’π’… πŸ’ Assign the coloring as follows: β€’ 𝜎(𝑒4π‘˜, 𝑒4π‘˜βˆ’1, 𝑒4π‘˜+1, 𝑣4π‘˜) = 𝑐2π‘˜ β€’ 𝜎(𝑣4π‘˜βˆ’2, 𝑣4π‘˜βˆ’3, 𝑣4π‘˜βˆ’1, 𝑒4π‘˜βˆ’2) = 𝑐2π‘˜βˆ’1 β€’ 𝜎(𝑒1, π‘’π‘š) = 𝑐 ⌈ π‘š 2 βŒ‰ To Prove: π‘§βˆ—[π‘Œπ‘š] < ⌈ π‘š 2 βŒ‰; π‘§βˆ—[π‘Œπ‘š] exists (π‘š βˆ’ 12) βˆ’ 𝐾1,3 stars, each color class is colored with one color. Hence, there exists an independent set remains uncolored we need one more color to complete the graph. Therefore, assumption is contradictory. Hence π‘ž[π‘Œπ‘š] = ⌈ π‘š 2 βŒ‰ . 3.2: Star Cochromatic Number of Line Graph of Prism Graphs Theorem 3.2.1: The Star Chromatic Number of Line Graph of Prism Graph π‘ž[𝐿(π‘Œπ‘š)] for π‘š β‰₯ 3 is π‘ž[𝐿(π‘Œπ‘š)] = ⌈ π‘š+2 2 βŒ‰ Proof: Let π‘Œπ‘š be prism graph with 2m vertices and 3m edges and 𝑉[π‘Œπ‘š] = {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š} and [𝐿(π‘Œπ‘š)] be line graph of prism graph with 3m vertices and 6m edges where 𝑉[𝐿(π‘Œπ‘š)] = {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑀𝑛: 1 ≀ 𝑛 ≀ π‘š}. Consider the color class 𝐢 = {𝑐1, 𝑐2, 𝑐3, … , 𝑐 ⌈ π‘š+2 2 βŒ‰ }. Define mapping 𝜎: {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑀𝑛: 1 ≀ 𝑛 ≀ π‘š} β†’ π‘π‘˜ βˆ€ π‘˜ = 1,2,3, … Case-(i): π’Ž = 𝒆𝒗𝒆𝒏 Assign the coloring as follows: β€’ 𝜎(𝑒4π‘˜βˆ’3, 𝑣4π‘˜βˆ’3, 𝑣4π‘˜βˆ’2, 𝑀4π‘˜βˆ’3, 𝑀4π‘˜βˆ’2) = 𝑐2π‘˜βˆ’1 β€’ 𝜎(𝑒4π‘˜βˆ’1, 𝑣4π‘˜βˆ’1, 𝑣4π‘˜ , 𝑀4π‘˜βˆ’1, 𝑀4π‘˜) = 𝑐2π‘˜ β€’ 𝜎(𝑒2π‘˜) = 𝑐 ⌈ π‘š+2 2 βŒ‰ Case-(ii): π’Ž = 𝒐𝒅𝒅 β€’ 𝜎(𝑒4π‘˜βˆ’2, 𝑣4π‘˜βˆ’3, 𝑣4π‘˜βˆ’2, 𝑀4π‘˜βˆ’3, 𝑀4π‘˜βˆ’2) = 𝑐2π‘˜βˆ’1 β€’ 𝜎(𝑒4π‘˜, 𝑣4π‘˜βˆ’1, 𝑣4π‘˜, 𝑀4π‘˜βˆ’1, 𝑀4π‘˜) = 𝑐2π‘˜ β€’ 𝜎(𝑒2π‘˜βˆ’1) = 𝑐 ⌈ π‘š+2 2 βŒ‰βˆ’1 β€’ 𝜎(π‘£π‘š, π‘€π‘š) = 𝑐 ⌈ π‘š+2 2 βŒ‰ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 255 https://internationalpubls.com To Prove: π‘§βˆ—[𝐿(π‘Œπ‘š)] < ⌈ π‘š+2 2 βŒ‰, say ⌈ π‘š+2 2 βŒ‰ βˆ’ 2; π‘§βˆ—[𝐿(π‘Œπ‘š)] exists ( π‘š 2 ) βˆ’ 𝐾1,3 stars, each color class is colored with one color. Hence, there exists an independent set and clique remains uncolored we need two more colors to complete the graph. Therefore, assumption is contradictory. Hence π‘§βˆ—[𝐿(π‘Œπ‘š)] = ⌈ π‘š+2 2 βŒ‰. 3.3: Star Cochromatic Number of Middle Graph of Prism Graphs Theorem 3.3.1: The Star Chromatic Number of Middle Graph of Prism Graph π‘ž[𝑀(π‘Œπ‘š)] for π‘š ≑ 0 π‘šπ‘œπ‘‘ 2 is π‘ž[π‘Œπ‘š] = ⌈ π‘š+4 2 βŒ‰ Proof: Let π‘Œπ‘š be prism graph with 2m vertices and 3m edges and 𝑉[π‘Œπ‘š] = {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š} and [𝑀(π‘Œπ‘š)] be middle graph of prism graph where subdividing each edge exactly once and join the adjacent vertices, the vertex set of 𝑉[𝑀(π‘Œπ‘š)] = {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑣𝑛 β€² : 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛 β€² : 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑀𝑛 β€² : 1 ≀ 𝑛 ≀ π‘š}. Consider the color class 𝐢 = {𝑐1, 𝑐2, 𝑐3, … , 𝑐 ⌈ π‘š+4 2 βŒ‰ }. Define mapping 𝜎: {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑣𝑛 β€² : 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛 β€² : 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑀𝑛 β€² : 1 ≀ 𝑛 ≀ π‘š} β†’ π‘π‘˜ βˆ€ π‘˜ = 1,2,3, … Assign the coloring as follows: β€’ 𝜎(𝑀4π‘˜βˆ’3 β€² , 𝑣4π‘˜βˆ’3 β€² , 𝑣4π‘˜βˆ’2 β€² , 𝑒4π‘˜βˆ’3 β€² , 𝑣4π‘˜βˆ’2 β€² , 𝑣4π‘˜βˆ’2, 𝑒4π‘˜βˆ’2) = 𝑐2π‘˜βˆ’1 β€’ 𝜎(𝑀4π‘˜βˆ’1 β€² , 𝑣4π‘˜βˆ’1 β€² , 𝑣4π‘˜ β€² , 𝑒4π‘˜βˆ’1 β€² , 𝑒4π‘˜ β€² , 𝑒4π‘˜ , 𝑣4π‘˜) = 𝑐2π‘˜ β€’ 𝜎(𝑒2π‘˜βˆ’1, 𝑣2π‘˜βˆ’1) = 𝑐 ⌈ π‘š+4 2 βŒ‰βˆ’1 β€’ 𝜎(𝑀2π‘˜ β€² ) = 𝑐 ⌈ π‘š+4 2 βŒ‰ To Prove: π‘§βˆ—[𝑀(π‘Œπ‘š)] < ⌈ π‘š+4 2 βŒ‰, say ⌈ π‘š+4 2 βŒ‰ βˆ’ 1; π‘§βˆ—[𝑀(π‘Œπ‘š)] exists ( π‘š 2 ) βˆ’ 𝐾1,3 stars, is colored with π‘š 2 colors. Hence, there exists an independent set remains uncolored we need two more colors to complete the graph. Therefore, assumption is contradictory. Hence π‘§βˆ—[𝑀(π‘Œπ‘š)] = ⌈ π‘š+4 2 βŒ‰. 4.1: Star Cochromatic Number of Sunlet Graphs Theorem 4.1.1: The Star Chromatic Number of Sunlet Graph π‘ž[π‘†π‘š] for π‘š β‰₯ 3 is π‘ž[π‘†π‘š] = ⌈ π‘š+3 3 βŒ‰ Proof: Let π‘†π‘š be the sunlet graph with 2m vertices and 3m edges and 𝑉[π‘†π‘š] = {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š}. Consider the color class 𝐢 = {𝑐1, 𝑐2, 𝑐3, … , 𝑐 ⌈ π‘š+3 3 βŒ‰ }. Define mapping 𝜎: {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š} β†’ π‘π‘˜ βˆ€ π‘˜ = 1,2,3, … Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 256 https://internationalpubls.com Case-(i): π’Ž ≑ 𝟎 π’Žπ’π’… πŸ‘ β€’ 𝜎(𝑣6π‘˜βˆ’4, 𝑣6π‘˜βˆ’5, 𝑣6π‘˜βˆ’3, 𝑒6π‘˜βˆ’4) = 𝑐2π‘˜βˆ’1 β€’ 𝜎(𝑣6π‘˜βˆ’1, 𝑣6π‘˜βˆ’2, 𝑣6π‘˜, 𝑒6π‘˜βˆ’1) = 𝑐2π‘˜ β€’ 𝜎(𝑒3π‘˜βˆ’2, 𝑒3π‘˜) = 𝑐 ⌈ π‘š+3 3 βŒ‰ To Prove: π‘§βˆ—[π‘†π‘š] < ⌈ π‘š+3 3 βŒ‰; π‘§βˆ—[π‘†π‘š] exists 3π‘š βˆ’ 𝐾1,3 stars, each color class is colored with one color. There exists an independent set (𝑒3π‘˜βˆ’2, 𝑒3π‘˜) remains uncolored we need one more color to complete the graph. Therefore, assumption is contradictory. Hence π‘ž[π‘†π‘š] = ⌈ π‘š+3 3 βŒ‰ . Case-(ii): π’Ž ≑ 𝟏 π’Žπ’π’… πŸ‘ β€’ 𝜎(𝑣6π‘˜βˆ’4, 𝑣6π‘˜βˆ’5, 𝑣6π‘˜βˆ’3, 𝑒6π‘˜βˆ’4) = 𝑐2π‘˜βˆ’1 β€’ 𝜎(𝑣6π‘˜βˆ’1, 𝑣6π‘˜βˆ’2, 𝑣6π‘˜, 𝑒6π‘˜βˆ’1) = 𝑐2π‘˜ β€’ 𝜎(π‘£π‘š, π‘’π‘š) = 𝑐 ⌈ π‘š+3 3 βŒ‰βˆ’1 β€’ 𝜎(𝑒3π‘˜βˆ’2, 𝑒3π‘˜) = 𝑐 ⌈ π‘š+3 3 βŒ‰ Case-(iii): π’Ž ≑ 𝟐 π’Žπ’π’… πŸ‘ β€’ 𝜎(𝑣6π‘˜βˆ’4, 𝑣6π‘˜βˆ’5, 𝑣6π‘˜βˆ’3, 𝑒6π‘˜βˆ’4) = 𝑐2π‘˜βˆ’1 β€’ 𝜎(𝑣6π‘˜βˆ’1, 𝑣6π‘˜βˆ’2, 𝑣6π‘˜, 𝑒6π‘˜βˆ’1) = 𝑐2π‘˜ β€’ 𝜎(π‘£π‘š, π‘£π‘šβˆ’1) = 𝑐 ⌈ π‘š+3 3 βŒ‰βˆ’1 β€’ 𝜎(𝑒3π‘˜βˆ’2, 𝑒3π‘˜, π‘’π‘š) = 𝑐 ⌈ π‘š+3 3 βŒ‰ To Prove: π‘§βˆ—[π‘†π‘š] < ⌈ π‘š+3 3 βŒ‰; π‘§βˆ—[π‘†π‘š] exists 3π‘š βˆ’ 𝐾1,3 stars, each color class is colored with one color. Hence, there exists an independent set and clique remains uncolored we need two more colors to complete the graph. Assumption is contradictory. Hence π‘ž[π‘Œπ‘š] = ⌈ π‘š+3 3 βŒ‰ . 4.2: Star Cochromatic Number of Line Graph of Sunlet Graphs Theorem 4.2.1: The Star Chromatic Number of Line Graph of Sunlet Graph π‘ž[𝐿(π‘†π‘š)] for π‘š β‰₯ 4 is π‘ž[π‘Œπ‘š] = ⌈ π‘š 2 βŒ‰ Proof: Let π‘†π‘š be sunlet graph with 2m vertices and 3m edges and 𝑉[π‘†π‘š] = {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š} and [𝐿(π‘†π‘š)] be line graph of prism graph with 3m vertices where 𝑉[𝐿(π‘†π‘š)] = {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š}. Consider the color class 𝐢 = {𝑐1, 𝑐2, 𝑐3, … , 𝑐 ⌈ π‘š 2 βŒ‰ }. Define mapping 𝜎: {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š} β†’ π‘π‘˜ βˆ€ π‘˜ = 1,2,3, … Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 257 https://internationalpubls.com Case-(i): π’Ž ≑ 𝟎 π’Žπ’π’… πŸ‘ β€’ 𝜎(𝑣6π‘˜βˆ’4, 𝑣6π‘˜βˆ’5, 𝑣6π‘˜βˆ’3, 𝑣6π‘˜βˆ’5, 𝑒6π‘˜βˆ’4) = 𝑐2π‘˜βˆ’1 β€’ 𝜎(𝑣6π‘˜βˆ’1, 𝑣6π‘˜βˆ’2, 𝑣6π‘˜, 𝑣6π‘˜βˆ’2, 𝑒6π‘˜βˆ’1) = 𝑐2π‘˜ β€’ 𝜎(𝑒3π‘˜) = 𝑐 ⌈ π‘š 2 βŒ‰ Case-(ii): π’Ž ≑ 𝟏 π’Žπ’π’… πŸ‘ β€’ 𝜎(𝑣6π‘˜βˆ’4, 𝑣6π‘˜βˆ’5, 𝑣6π‘˜βˆ’3, 𝑣6π‘˜βˆ’5, 𝑒6π‘˜βˆ’4) = 𝑐2π‘˜βˆ’1 β€’ 𝜎(𝑣6π‘˜βˆ’1, 𝑣6π‘˜βˆ’2, 𝑣6π‘˜, 𝑣6π‘˜βˆ’2, 𝑒6π‘˜βˆ’1) = 𝑐2π‘˜ β€’ 𝜎(π‘£π‘š, π‘’π‘š) = 𝑐 ⌈ π‘š 2 βŒ‰βˆ’1 β€’ 𝜎(𝑒3π‘˜) = 𝑐 ⌈ π‘š 2 βŒ‰ Case-(iii): π’Ž ≑ 𝟐 π’Žπ’π’… πŸ‘ β€’ 𝜎(𝑣6π‘˜βˆ’4, 𝑣6π‘˜βˆ’5, 𝑣6π‘˜βˆ’3, 𝑣6π‘˜βˆ’5, 𝑒6π‘˜βˆ’4) = 𝑐2π‘˜βˆ’1 β€’ 𝜎(𝑣6π‘˜βˆ’1, 𝑣6π‘˜βˆ’2, 𝑣6π‘˜, 𝑣6π‘˜βˆ’2, 𝑒6π‘˜βˆ’1) = 𝑐2π‘˜ β€’ 𝜎(π‘£π‘š, π‘£π‘šβˆ’1) = 𝑐 ⌈ π‘š 2 βŒ‰βˆ’1 β€’ 𝜎(𝑒3π‘˜, π‘’π‘š , π‘’π‘šβˆ’1) = 𝑐 ⌈ π‘š 2 βŒ‰ To Prove: π‘§βˆ—[𝐿(π‘†π‘š)] < ⌈ π‘š 2 βŒ‰, say ⌈ π‘š 2 βŒ‰ βˆ’ 1; π‘§βˆ—[𝐿(π‘†π‘š)] exists ( π‘š 3 ) βˆ’ 𝐾1,3 stars, each color class is colored with π‘š 3 colors. Hence, there exists an independent set remains uncolored we need one more colors to complete the graph. Therefore, assumption is contradictory. Hence π‘§βˆ—[𝐿(π‘†π‘š)] = ⌈ π‘š 2 βŒ‰. 4.3: Star Cochromatic Number of Middle Graph of Sunlet Graphs Theorem 4.3.1: The Star Chromatic Number of Middle Graph of Sunlet Graph π‘ž[𝑀(π‘†π‘š)] for π‘š ≑ 0 π‘šπ‘œπ‘‘ 3 is π‘ž[𝑀(π‘†π‘š)] = ⌈ π‘š+4 3 βŒ‰ Proof: Let π‘†π‘š be sunlet graph with 2m vertices and 3m edges and 𝑉[π‘†π‘š] = {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š} and [𝑀(π‘†π‘š)] be middle graph of sunlet graph where subdividing each edge exactly once and join the adjacent vertices, the vertex set of 𝑉[𝑀(π‘†π‘š)] = {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑣𝑛 β€² : 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛 β€² : 1 ≀ 𝑛 ≀ π‘š}. Consider the color class 𝐢 = {𝑐1, 𝑐2, 𝑐3, … , 𝑐 ⌈ π‘š+4 3 βŒ‰ }. Define mapping 𝜎: {𝑣𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑣𝑛 β€² : 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛: 1 ≀ 𝑛 ≀ π‘š} βˆͺ {𝑒𝑛 β€² : 1 ≀ 𝑛 ≀ π‘š} β†’ π‘π‘˜ βˆ€ π‘˜ = 1,2,3, … Assign the coloring as follows: β€’ 𝜎(𝑣6π‘˜βˆ’4 β€² , 𝑒6π‘›βˆ’4 β€² , 𝑒6π‘›βˆ’3 β€² , 𝑣6π‘˜βˆ’5 β€² , 𝑣6π‘˜βˆ’3 β€² , 𝑣6π‘˜βˆ’4, 𝑣6π‘˜βˆ’3) = 𝑐2π‘˜βˆ’1 β€’ 𝜎(𝑣6π‘˜βˆ’1 β€² , 𝑒6π‘›βˆ’1 β€² , 𝑒6𝑛 β€² , 𝑣6π‘˜βˆ’2 β€² , 𝑣6π‘˜ β€² , 𝑣6π‘˜βˆ’1, 𝑣6π‘˜) = 𝑐2π‘˜ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 258 https://internationalpubls.com β€’ 𝜎(π‘’π‘˜, 𝑣3π‘˜βˆ’2) = 𝑐 ⌈ π‘š+4 3 βŒ‰ To Prove: π‘§βˆ—[𝑀(π‘†π‘š)] < ⌈ π‘š+4 3 βŒ‰, say ⌈ π‘š+4 3 βŒ‰ βˆ’ 1; π‘§βˆ—[𝑀(π‘†π‘š)] exists ( π‘š 3 ) βˆ’ 𝐾1,3 stars, is colored with π‘š 3 colors. Hence, there exists an independent set remains uncolored we need one more color to complete the graph. Assumption is contradiction. Hence π‘§βˆ—[𝑀(π‘†π‘š)] = ⌈ π‘š+4 3 βŒ‰. 5. Conclusion: In this article, we examined the star cochromatic number of prism graphs, the line graph of prism graphs, the middle graph of prism graphs, the star cochromatic index for sunlet graphs, the line graph of sunlet graphs and the middle graph of sunlet graphs. Further improvements will encompass more central, line, total and middle graphs in various graph families. References [1] Hassler Whitney, Congruent graphs and the connectivity of graphs, Amer. J. Math. 1932. [2] Kowsalya. V, Vernold Vivin. J and Venkatachalam. M, On Star Chromatic Number of Sunlet Graph Families, Ars Combinatoria, Vol-123 (431-437), 2015. [3] Lesinak and Straight, A Cochromatic Number of a Graph, 1977. [4] M. Poobalaranjani, A Study on Cocoloring and Variants of Cocoloring of Graphs, Ph. D Thesis, 2020 [5] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas, The strong perfect graph theorem, Ann. of Math, 2006. [6] T. Hamada and I. Yoshimura, Traversability and connectivity of the Middle Graphs of a Graph, Discrete Mathematics, 14, 1976. [7] Vernold Vivin. J, Kowsalya. V, and Vimal Kumar. S, On Star Chromatic Number of Prism Graph Families, TWMS Journal of Applied and Engineering Mathematics, Vol.9, No.3 687- 692, 2019. https://shodhganga.inflibnet.ac.in/jspui/handle/10603/466685