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16 Prime Labeling of Triangular Book and Cycle International Journal of Scientific Research and Modern Education (IJSRME) Impact Factor: 6.225, ISSN (Online): 2455 – 5630 (www.rdmodernresearch.com) Volume 2, Issue 2, 2017 PRIME LABELING OF TRIANGULAR BOOK AND CYCLE- CACTUS GRAPHS V. Ganesan*, S. Mahalakshmi** & M. P. Sathya** * PG and Research of Mathematics, Thiru Kolanjiappar Government Arts College, Vridhachalam, Tamilnadu ** PG and Research of Mathematics, Sri Vinayaga College of Arts and Science, Ulundurpet, Tamilnadu Cite This Article: V. Ganesan, S. Mahalakshmi & M. P. Sathya, “Prime Labeling of Triangular Book and Cycle-Cactus Graphs”, International Journal of Scientific Research and Modern Education, Volume 2, Issue 2, Page Number 16-20, 2017. Copy Right: © IJSRME, 2017 (All Rights Reserved). This is an Open Access Article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract: (푛) In this paper, we prove that the Triangular book 퐵푛 and Cycle-Cactus 퐶푘 graphs are prime graphs. Here, we investigate and prove that new results for Triangular book 퐵푛 admits prime labeling when 푛 is even (푛) and odd separately. We also show that the Cycle-Cactus 퐶푘 admits prime labeling for all 푛 and 푘 where 푘 ≥ 3. Key Words: Prime Labeling and Prime Graphs, Triangular Book & Cycle-Cactus 1. Introduction: We consider only simple, finite, connected and undirected graph 퐺 = (푉 퐺 , 퐸 퐺 ) with 푝 vertices and 푞 edges. For standard terminology and notations, we follow Bondy and Murty [1]. We will provide brief summary of definitions and other information which are necessary for the present investigations. Definition 1.1: An independent set of vertices in a graph 퐺 is a set of mutually non-adjacent vertices. Definition 1.2: If the vertices are assigned values subject to certain conditions then it is known as graph labeling. Definition 1.3: Let 퐺 = (푉, 퐸) be a graph with 푛 vertices. A bijection 푓: 푉 퐺 → {1,2, . , 푛} is called a prime labeling if for each edge 푒 = 푢푣, gcd 푓 푢 , 푓 푣 = 1. A graph which admits prime labeling is called a prime graph. Definition 1.4: A Triangular book 퐵푛 is defined as 퐵푛 = 퐾2 + 퐾 푡 Definition 1.5: The Cycle-Cactus consisting of 푛 copies of cycle 푐푘 , k≥3 concatenated at exactly one vertex is (푛) denoted as 퐶푘 . Definition 1.6: The set of vertices adjacent to a vertex 푢 of 퐺 is denoted by 푁(푢). Definition 1.7: (Bertrand’s Postulate) For every 푛 ≥ 2, there exists a prime 푝 such that 푛 < 푝 < 2푛 The notion of prime labeling was originated by Entringer and it was discussed by Tout et al [8].Fu and Huang [4] proved that 푃푛 and 퐾1,푛 are prime graphs. Lee et al [7] proved that 푊푛 is a prime graph if and only if 푛 is even. Deretsky et al [3] proved that 퐶푛 is a prime graph. In [6], Ganesan and Balamurugan have proved that the Mongolian tent graph is a prime graph. In [9], Vaidya and Kanmani proved the prime labeling for some cycle related graphs. Carlson in [2] has proved that the Generalised Books and 퐶푚 snakes are prime graphs. Remark Using Bertrand’s principle, we can find the value of the prime number 푝 such that either 푝 = 2푛 − 1 or 푝 = 2푛 + 1 or both 2. Main Results: Theorem 2.1: The triangular book 퐵푛 = 퐾2 + 퐾 푡 admits prime labeling when 푛 is even. Proof: Let 퐵푛 = 퐾2 + 퐾 푡 be a triangular book where 푡 is even. Let 푢 and 푣 be the vertices of 퐾2 and 푢1,푢2, … , 푢푡 be the vertices of 퐾푡 Then 푉 퐵푛 = 푡 + 2 Define a function 푓: 푉 퐵푛 → {1,2, . , 푡 + 2} as follows For labeling the vertices of 퐵푛 ,we consider the following two cases Case:1 When 푡 is even and 푡 + 1 is a prime number Let 푓 푢 = 1 and 푓 푣 = 푡 + 1 푓 푢푖 = 푖 + 1, for 1 ≤ 푖 ≤ 푡 − 1 and 푓 푢푡 = 푡 + 2 Case:2 When 푡 is even and 푡 + 1 is not a prime number Let 푓 푢 = 1 and 푓 푣 = 푡 − 1 푓 푢푖 = 푖 + 1, for 1 ≤ 푖 ≤ 푡 − 3 푓 푢푡−2 = 푡 16 International Journal of Scientific Research and Modern Education (IJSRME) Impact Factor: 6.225, ISSN (Online): 2455 – 5630 (www.rdmodernresearch.com) Volume 2, Issue 2, 2017 푓 푢푡−1 = 푡 + 1 푓 푢푡 = 푡 + 2 In view of above defined labeling pattern, 푓 satisfy the condition of prime labeling Therefore, 퐵푛 admits prime labeling Hence, The triangular book 퐵푛 = 퐾2 + 퐾푡 is a prime graph. Illustrations Case 1: When 푡 is even and 푡 + 1 is a prime number (i) Let 푡 = 6 and 푡 + 1 = 7 (prime) Figure 2.1: Triangular book 퐵8 and its prime labeling (ii) Let 푡 = 10 and 푡 + 1 = 11 (prime) Figure 2.2: Triangular book 퐵12 and its prime labeling Case 2: When 푡 is even and 푡 + 1 is not a prime number (i) Let 푡 = 8 and 푡 + 1 = 9 is not a prime Figure 2.3: Triangular book 퐵10 and its prime labeling 17 International Journal of Scientific Research and Modern Education (IJSRME) Impact Factor: 6.225, ISSN (Online): 2455 – 5630 (www.rdmodernresearch.com) Volume 2, Issue 2, 2017 (ii) Let 푡 = 14 and 푡 + 1 = 15 is not a prime Figure 2.4: Triangular book 퐵16 and its prime labeling Theorem 2.2: The triangular book 퐵푛 = 퐾2 + 퐾 푡 admits prime labeling when 푡 is odd. Proof: Let 퐵푛 = 퐾2 + 퐾 푡 be a triangular book where 푡 is an odd integer Let 푢 and 푣 be the vertices of 퐾2 and 푢1,푢2, … , 푢푡 be the vertices of 퐾푡 Then 푉 퐵푛 = 푡 + 2= an odd number Define a function 푓: 푉 퐵푛 → {1,2, . , 푡 + 2} as follows we consider the following two cases Case:1 When 푡 is odd and 푡 is a prime number Let 푓 푢 = 1 and 푓 푣 = 푡 푓 푢푖 = 푖 + 1, for 1 ≤ 푖 ≤ 푡 − 2 and 푓 푢푡−1 = 푡 + 1 푓 푢푡 = 푡 + 2 Case:2 When 푡 is odd and 푡 is not a prime number Let 푓 푢 = 1 and 푓 푣 = 푡 + 2 푓 푢푖 = 푖 + 1, for 1 ≤ 푖 ≤ 푡 In view of above defined labeling pattern, 푓 satisfy the condition of prime labeling Hence, the triangular book 퐵푛 = 퐾2 + 퐾푡 is a prime graph when 푡 is odd Illustrations: Case 1: When 푡 is an odd integer and 푡 = 푝 (prime) Let 푡 = 7 and 푡 is prime number Figure 2.5: Triangular book 퐵9 and its prime labeling 18 International Journal of Scientific Research and Modern Education (IJSRME) Impact Factor: 6.225, ISSN (Online): 2455 – 5630 (www.rdmodernresearch.com) Volume 2, Issue 2, 2017 Case 2: When 푡 is an odd number and 푡 ≠ 푝 (prime) Let 푡 = 9 and 푡 ≠ 푝 Figure 2.6: Triangular book 퐵11 and its prime labeling Theorem 2.3: (푛) The graph Cycle-Cactus 퐶푘 admits prime labeling Proof: (푛) Let 퐺 = 퐶푘 be the cycle-cactus graph consisting of 푛 copies of the cycle 퐶푘 with 푘 vertices, 푘 ≥ 3concatenated at exactly one vertex. (1) (2) (푛) Let 퐶푘 , 퐶푘 ,… , 퐶푘 be the 푛 copies of the cycle 퐶푘 and these 푛 copies concatenated at the common (1) vertex 푣. Let 푣, 푣12 , 푣13, … , 푣1푘 be the vertices of the first copy 퐶푘 of the cycle 퐶푘 , 푣, 푣22, 푣23 , … , 푣2푘 be the (2) vertices of the second copy. 퐶푘 of the cycle 퐶푘 and finally,let 푣, 푣푛2, 푣푛3, … , 푣푛푘 be the vertices of the nth (푛) copy 퐶푘 of the cycle 퐶푘 . We assume that all the 푛 copies of the cycle 퐶푘 concatenated at the common vertex 푣. Then 푉 퐺 = 푛 푘 − 1 + 1 Define a function 푓: 푉 퐺 → {1,2, . , 푛 푘 − 1 + 1} as follows Let 푓 푣 = 1 푓 푣1푖 = 푖 + 1, for 2 ≤ 푖 ≤ 푘 푓 푣2푖 = 푘 − 1 + 푖, for 2 ≤ 푖 ≤ 푘 푓 푣3푖 = 2푘 − 2 + 푖, for 2 ≤ 푖 ≤ 푘 ⋮ ⋮ 푓 푣푛푖 = 푛 − 1 푘 − 푛 − 1 + 푖, for 2 ≤ 푖 ≤ 푘 In view of above defined labeling pattern, 푓 satisfy the condition of prime labeling (푛) Hence, the Cycle-Cactus 퐶푘 admits prime labeling. Illustrations: (1) Let 푘 = 5 and 푛 = 3 (3) Figure 2.7: The Cycle-Cactus 퐶5 and its prime labeling 19 International Journal of Scientific Research and Modern Education (IJSRME) Impact Factor: 6.225, ISSN (Online): 2455 – 5630 (www.rdmodernresearch.com) Volume 2, Issue 2, 2017 (2) Let 푘 = 4 and 푛 = 4 (4) Figure 2.8: The Cycle-Cactus 퐶4 and its Prime Labeling 3. Conclusion: Labeling of graph is a potential area of research due to its diversified applications and it is very interesting to investigate whether any graph (or) graph family admits a particular labeling or not? Here, we contribute three results in the context of prime labeling of two graphs, namely triangular book and cycle-cactus. Analogous results can be investigated for various graphs and similar results can be obtained in the context of different graph labeling techniques. 4. References: 1. J. A. Bondy and U. S. R. Murty, “Graph Theory and applications”, (North Holland), New York(1976). 2. K. Carlson, “Generalised Books and Cm snakes are prime graphs”, Ars Combina., 80(2006) 215-221. 3. T. O. Deretsky, S. M. Lee and J. Mitchem,” On vertex Prime labeling of graphs in graph theory”, Combinatories and applications, Vol.
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