Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 991 https://internationalpubls.com Chromatic Number of Split Graph of Cycle Cn G.Prabhakaran1,K.Balasangu2, S.Vijayaraj3 ,V.Ganesan4 1Assistant professor, Department of Mathematics, Sri Vinayaga College of Arts and Science. Ulundurpet 2Associate professor, Department of Mathematics, T.K. Govt. Arts College. Vridhachalam 3Assistant professor, Department of Mathematics, Govt.Arts and Science College. Kallakurichi 4Assistant professor, Department of Mathematics, T.K. Govt. Arts College. Vridhachalam. 1Email-igprabhakaran19@gmail.com 2Email-balasangu76 @gmail.com 3Email-vijayaraj90@gmail.com 4Email-vganesanmath@gmail.com Article History: Received: 04-07-2025 Revised: 23-08-2025 Accepted: 26-09-2025 Abstract: In this paper we have discuss about how to find chromatic number of split graph and discussed about how to assign the colours. Also discussed about how to label vertices. The application of various graph coloring approaches in domains such as automated differentiation, mobile networks, optical networks, medical data mining, game theory, and radio networks. Definition1.1 Coloring In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring Definition1.2 Split graph A split graph is a graph whose vertices can be partitioned into a clique and an independent vertex set. Definition1.3 Chromatic number The chromatic number of a graph is the minimal number of colours needed to colour the vertices in such a way that no two adjacent vertices have the same colour Definition1.4 Chromatic number for split graph mailto:1Email-igprabhakaran19@gmail.com mailto:Email-vganesanmath@gmail.com mailto:3Email-vijayaraj90@gmail.com mailto:Email-vganesanmath@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 992 https://internationalpubls.com Based on the result, to determine the locating Based on the result, to determine the locating chromatic number for split graph of cycle, by dividing two cases. The first case when odd 𝑛 and second case when even 𝑛. So that, obtained the locating chromatic number for split graph of cycle = 4 for odd 𝑛 and 5 for even 𝑛. of cycle, by dividing two cases. The first case when odd 𝑛 and second case when even 𝑛. So that, obtained the locating chromatic number for split graph of cycle 𝐢𝑛 = 4 for odd and 5 for even 𝑛. Definition1.5 Cycle graph A cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with n vertices is called Cn. The number of vertices in Cn equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it. Definition1.6 Path graph A Path graph, also known as a linear graph, is a graph where the vertices are arranged in a line with edges connecting vertices that are adjacent Definition1.7 Middle graph The Middle graph M(G) of a graph G is the graph in which the vertex set is V (G) βˆͺ E(G) and two vertices are adjacent if and only if either they are adjacent edges of G or one is vertex of G and the other is an edge incident with it. Algorithm Finding the chromatic number of split graph of cycle cn Step:1 Let spl(Cn ) be the split graph of cycle graph cn Step:2 Label the vertices of spl (Cn ) by using the known algorithm which is a prime graph whose vertices are 𝑃1,𝑃2,…..π‘ƒπ‘š, and 𝑃1 β€² ,𝑃2 β€² , … . 𝑃𝑛 β€² Step:3 Assign the Colour , say Red to the vertices of the spl (Cn ) having even label. therefore we assign the colour Red to the verticxes 𝑃1 β€² ,𝑃2 β€² , … . 𝑃𝑛 β€² Step:4 The remaining odd labels of spl(Cn ) are 𝑣1 ,𝑣2 ……𝑣𝑛 Step:5 Assign the colour to the odd label vertices, fix any vertex say P1 and assign any colour other than Red, therefore we assign the colour Green to P1. next for the vertex P2, The adjacent vertices are 𝑃1 β€² , 𝑃3 β€² ,𝑃1,𝑃3, But we already we assigned the colour red to 𝑃1 β€² , 𝑃3 β€² and green to 𝑃1 https://en.wikipedia.org/wiki/Graph_(discrete_mathematics) https://en.wikipedia.org/wiki/Cycle_(graph_theory) https://en.wikipedia.org/wiki/Vertex_(graph_theory) https://en.wikipedia.org/wiki/Simple_graph https://en.wikipedia.org/wiki/Edge_(graph_theory) https://en.wikipedia.org/wiki/Degree_(graph_theory) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 993 https://internationalpubls.com Therefore we should assign different colour other than Red and Green therefore we shall assign Yellow to 𝑃2, Step:6 Repeat step-5 to all the vertices 𝑝1 , , 𝑝2 β€² , 𝑝3 , … . . , 𝑝𝑛 β€² untill all the vertices have been colourd Therefore cr(spl(cn)) = 3 Illustration -1.1 Fig. 1.1 split graph of C8 Illustration -1.2 Fig:1.2.Labelled spl (C8 ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 994 https://internationalpubls.com Illustration-1.3 Fig: 1.3 Colouring Red for even labeled vertices Illustration -1.4 Fig: 1.4 colouring different colours for adjacent vertices P1 (Green ) P2 (Yellow) Illustration -1.5 Fig: 1.5 Colouring Green for P3 and Yellow for P4 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 995 https://internationalpubls.com Illustration -1.6 Fig:1.6 Colouring Green for P5 and Yellow for P6 Illustration -1.7 Fig:1.7 Colouring Green for P5and Yellow for P6 Illustration -1.8 Fig:1.8 Colouring Green for P7 and Yellow for P8 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 996 https://internationalpubls.com Chromatic number of middle graph of path graph Pn Algorithm Finding the chromatic number of middle graph of path graph Pn Step:1 Let MI(Pn) be the middle graph of path graph Pn Step:2 We have to label the vertices of middle graph of path graph whose vertices are P1,P2,P3….Pn and P1’,P2’,….Pn’ Step:3 Assign any colour say red to the vertices of the MI(Pn) having even labels. Therefore we assign the colour red to the vertices P1,P2,….Pn Step:4 The remaining odd labels of MI(Pn) are P1’,P2’,….Pn’ Step:5 Fix any vertex say P1’ and assign any colour other than red therefore we assign the colour green to P1’. Next for the vertex P2’ The adjacent vertices are P2,P1’,P3,P3’ but already we assigned the colour red to P2,P3 and green to P1’ Therefore we should assign different colour other than red and green, therefore we shall assign yellow to P2’,P4’ Step:6 Repeat step-5 to all the vertices P1’,P2’,…..Pn’ until all the vertices have been coloured Therefore Cr(MI(Pn))=3 Illustration-2.1 Figure 2.1 middle graph P5 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 997 https://internationalpubls.com Illustration-2.2 Figure 2.2 Labeled MI(P5 ) Illustration-2.3 Figure 2.3 Colouring for even labeled vertices Illustration-2.4 Figure 2.4 Colouring different colour for adjacent vertices References [1] Gallian, J. A. (2022). A Dynamic Survey of Graph Labeling. Electronic Journal of Combinatorics, 6(25), 4-623. Article DS6. [2] S. Lavanya and V.Ganesan Prime Labeling of Split Graph of Coconut Tree CT(m,n), International Journal of Advanced Scientific Research and Management, Volume 5 Issue 4, Apr (2020), ISSN 2455-6378 14 www. ijasrm.com. [3] A. Uma Maheswari and S. Purnalakshimi, Engineering Arithmetic Number Labelling for Banana tree, Olive tree, Shrub, Jelly fish, Tadpole graphs. International Journal of Mechanical. ISSN: 0974-5823 Vol. 7 (Special Issue, Jan.-Mar. 2022). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 998 https://internationalpubls.com [4] Dian KastikaSyofyan, Edy Tri Baskoro, Hilda Assiyatun, The Locating-Chromatic Number of Binary Trees. International Conference on Graph Theory and Information Security. Procedia Computer Science 74, 79 – 83 (2015). www.sciencedirect.com. [5] P. Sumathi, S. Tamilselvi, Modular Chromatic Number of Certain Cyclic Graphs. Journal of Algebraic Statistics, Volume 13, No. 3, p. 4459- 4466 (2022). [6] P. Sumathi, S. Tamilselvi, Modular coloring on inflated graphs. Industrial Engineering Journal, ISSN 0970-2555 Volume 52, Issue 4, NO. 2, April (2023). [7] P. Sumathi, S. Tamilselvi, The modular chromatic number of the corona product of a generalized Jahangir graph, Indian Journal of Science and Technology, Volume: 16, Issue: 46, Pages: 4309- 4327 (2023), DOI: 10.17485/IJST/v16i46.2028. http://www.sciencedirect.com/