International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-8 Issue-5, January 2020 Some Results on 2-odd Labeling of Graphs P. Abirami, A. Parthiban, N. Srinivasan Abstract: A graph is said to be a 2-odd graph if the vertices of II. MAIN RESULTS can be labelled with integers (necessarily distinct) such that for In this section, we establish 2-odd labeling of some graphs any two vertices which are adjacent, then the modulus difference of their labels is either an odd integer or exactly 2. In this paper, such as wheel graph, fan graph, generalized butterfly graph, we investigate 2-odd labeling of some classes of graphs. the line graph of sunlet graph, and shell graph. Lemma 1. Keywords: Distance Graphs, 2-odd Graphs. If a graph has a subgraph that does not admit a 2-odd I. INTRODUCTION labeling then G cannot have a 2-odd labeling. One can note that the Lemma 1 clearly holds since if has All graphs discussed in this article are connected, simple, a 2-odd labeling then this will yield a 2-odd labeling for any subgraph of . non-trivial, finite, and undirected. By , or simply , Definition 1: [5] we mean a graph whose vertex set is and edge set is . The wheel graph is a graph with vertices According to J. D. Laison et al. [2] a graph is 2-odd if there , formed by joining the central vertex to all the exists a one-to-one labeling (“the set of all vertices of . integers”) such that for any two vertices and which are Theorem 1: adjacent, the integer is either exactly 2 or an The wheel graph admits a 2-odd labeling for all . odd integer. They also defined that Proof. and called a 2-odd labeling of . So is a 2-odd graph if Let be the given wheel graph on vertices. It is clear and only if there exists a 2-odd labeling of . One can notice to see that there are vertices on the rim, namely that in a 2-odd labeling, the vertex labels of are distinct, but and the central vertex, say . Now define an the edge labels are not so. Moreover, by this injective function as follows: without loss of definition, may still be either 2 or odd if is not an generality, let , , , and edge of . The stronger version of 2-odd labeling is known as for . An easy check shows that a prime distance labeling of graphs. For a detailed study on is the required 2-odd labeling of as prime distance graphs one can refer to [4, 5, 6, 7, and 8]. In are all odd numbers for 3≤ ≤ , 0− 1=2, this present paper, we investigate 2-odd labeling of certain , for classes of graphs. , and is an odd integer. Note 1: Definition 2: [5] A 2-odd labeling of a graph is not unique. A fan graph is defined as the graph , where Example 1: is the empty graph on one vertex and is a path on A path can be labeled with or . vertices. Theorem 2: The fan graph admits a 2-odd labeling for any . Proof. Let be the given fan graph on vertices. We label the vertex as and vertices of a path as . One can note that Now define an injective labeling as follows: without Revised Manuscript Received on January 31, 2020. * Correspondence Author loss of generality, let and for P. Abirami, Research Scholar, Department of Mathematics, St. Peter's . An easy check shows that is the desired 2-odd Institute of Higher Education and Research, Avadi, Chennai-600 054, Tamil Nadu, India. Email: [email protected] labeling of . Dr. A. Parthiban*, Assistant Professor, Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional University, Phagwara-144 411, Punjab, India. Email: [email protected] Dr. N. Srinivasan, Professor and Head, Department of Mathematics, St. Peter's Institute of Higher Education and Research, Avadi, Chennai-600 054, Tamil Nadu, India. Email: [email protected] Published By: Retrieval Number: E7041018520/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.E7041.018520 5644 & Sciences Publication Some Results on 2-odd Labeling of Graphs The line graph of a graph , denoted by , is a graph whose vertices are the edges of , and if then if and share a vertex in . Theorem 4: The line graph of a sunlet graph is a 2-odd graph for . Proof. Let be the given sunlet graph on vertices. Obtain the line graph of a sunlet graph whose vertex set is defined as where and . Now define a one-to-one labeling as follows: without loss of generality, let for , Fig. 1. A 2-odd labeling of , for , and . One can see that is the required 2-odd labeling of . Fig. 2. A 2-odd labeling of Definition 3: [1] The generalized butterfly graph, denoted by , is a graph formed by adjoining vertices to every wing with the assumption that “sum of inserting vertices to every wing are Fig. 3. A 2-odd labeling of same.” Theorem 3: Definition 6: [3] A shell graph is defined as a cycle with The generalized butterfly graph is a 2-odd graph for . chords sharing a common end vertex called the apex. Proof. Theorem 5: The shell graph permits a 2-odd labeling for all Let be the given generalized butterfly graph. One can notice that and Proof. Let be and Let be the given shell graph with . We be label the vertices of as and let . We label the apex vertex as be the apex vertex. Define an injective labeling the vertices on right wing as , and the vertices as follows: without loss of generality, on left wing as . Define a one-one labeling let and for . as follows: without loss of generality, let Clearly is the required 2-odd labeling of . , for any and for . One can easily see that is the required 2-odd labeling of . Definition 4: [9] The sunlet graph is a graph on vertices obtained by attaching pendant edges to the cycle . Definition 5: [9] Published By: Retrieval Number: E7041018520/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.E7041.018520 5645 & Sciences Publication International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-8 Issue-5, January 2020 AUTHORS PROFILE Dr. A. Parthiban, born on June 03, 1988. He is working as an Assistant Professor in the Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional University, Punjab, India. He did his Under graduation, Post graduation, and Master of Philosophy at Loyola College, Chennai. He pursued his Doctor of Philosophy (Ph.D) at Madras Christian College, Chennai under the supervision of Prof. N. Gnanamalar David. He has also qualified SET. He is the awardee of UGC-SRF/JRF (RGNF) during 2015-2018. He has worked as JRF in a NBHM research project at Velammal Engineering College, Chennai during 2011-2013. His area of specialization is graph theory and his research area includes graph labeling, graph coloring, equitable power domination, algebraic graph theory etc. He has more than 25 publications in reputed journals/conference proceedings (SCIE/Scopus/UGC-CARE/Peer-reviewed). He has attended various international and national conferences/workshops across the world. He has supervised 2 PG and 3 UG students and currently supervising 4 Ph.D. scholars and 2 PG students. Fig. 4. A 2-odd labeling of P. Abirami, born on Nov 19th 1986. She is a research scholar and pursuing her Ph.D. under the supervision of III. CONCLUSION Prof. N. Srinivasan at the Department of Mathematics, St. Peter’s Institute of Higher Education and Research, The 2-odd labeling of various classes of graphs such as Chennai. She did her M.Phil in St. Peter’s Institute of wheel graph, fan graph, generalized butterfly graph, line Higher Education and Research, Chennai. She has graph of sunlet graph, and shell graph are established. completed her Post Graduation and Under Graduation in Shanmuga Industries Arts and Science College, Thiruvannamalai. Her area of research Investigating 2-odd labeling of other classes of graphs is still is graph labeling. open and this is for future work. One can also explore the exclusive applications of 2-odd labeling in real life problems. Prof. N. Srinivasan, is the head of the Department of Mathematics, St. Peter’s Institute of Higher ACKNOWLEDGMENT Education and Research, Chennai. His area of research includes graph theory. He has a very good Authors express their sincere gratitude to the referees of this number of research publications in reputed journals and conference proceedings. article for their valuable comments and suggestions for the refinement of this article performance. REFERENCES 1. Hafidhyah Dwi Wahyuna and Diari Indriati, “On the total edge irregularity strength of generalized butterfly graph”, Journal of Physics: Conf. Series, vol. 1008, pp. 1-6, 2018. 2. J. D. Laison, C. Starr, and A. Walker, “Finite prime distance graphs and 2-odd graphs”, Discrete Mathematics, vol. 313, pp. 2281-2291, 2013. 3. S. Meena, M. Renugha, and M. Sivasakthi, “Cordial labeling for different types of shell graph”, International Journal of Scientific & Engineering Research, vol. 6 (9), pp. 1282-1288, 2015. 4. A. Parthiban and N. Gnanamalar David, “On prime distance labeling of graphs”, Lecture Notes in Computer Science, vol. 31, pp. 238-241, 2017. 5. A. Parthiban and N. Gnanamalar David, “On finite prime distance graphs”, Indian Journal of Pure and Applied Mathematics, 2017. (To appear) 6. A. Parthiban and N. Gnanamalar David, “On prime distance labeling of some special graphs”, Contemporary Studies in Discrete Mathematics, vol.