European Journal of Molecular & Clinical Medicine ISSN 2515-8260 Volume 07, Issue 07, 2020
On 2-Odd Labeling Of Graphs
P. Abirami1, A. Parthiban2, N. Srinivasan3
1,3Department of Mathematics, St. Peter's Institute of Higher Education and Research, Avadi, Chennai - 600 054, Tamil Nadu, India 2Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional University, Phagwara-144 411, Punjab, India
[email protected], [email protected], [email protected]
Abstract: A graph 푮 is a 2-odd graph if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the absolute difference of their labels is either an odd integer or 2. In this paper, we investigate 2-odd labeling of various classes of graphs.
Keywords: Distance Graphs, 2-odd Graphs
1. Introduction
All the graphs considered in this paper are finite, simple, connected, and undirected. By 퐺(푉, 퐸), or simply 퐺, we mean a graph 퐺 with vertex set 푉 and edge set 퐸. According to Laison et al. [3] a graph 퐺 is 2-odd if there exists a one-to-one labeling ℎ: 푉(퐺) → 푍 (“the set of all integers”) such that for any two adjacent vertices 푢 and 푣, the integer |ℎ(푢) − ℎ(푣)| is either an odd integer or exactly 2. They also defined that ℎ(푢푣) = |ℎ(푢) − ℎ(푣)| and called ℎ a 2-odd labeling of 퐺. So 퐺 is a 2-odd graph if and only if there exists a 2-odd labeling of 퐺. Note that in a 2-odd labeling, the vertex labels of 퐺 must be distinct, but the edge labels need not be so. Furthermore, by this definition ℎ(푢푣) may still be either 2 or odd if 푢푣 is not an edge of 퐺. In this paper, we obtain 2-odd labeling of some graphs. Note 1: A 2-odd labeling of a graph 퐺 is not unique. Example 1: A path 푃푛 can be labeled with 5, 10, . . . ,5푛 or 2, 4, . . . , 2푛.
2. Main Results In this section, we establish the 2-odd labeling of certain graphs. We also make use of some well- known number theoretic concepts in proving some of the results. Lemma 1. [3] If a graph 퐺 has a subgraph that does not admit a 2-odd labeling then G cannot have a 2-odd labeling. One can notice that the Lemma 1 clearly holds since if 퐺 has a 2-odd labeling then this will yield a 2-odd labeling for any subgraph of 퐺. Theorem 1. [3] 1. Every bipartite graph is 2-odd. 2. Every cycle is 2-odd. Definition 1. [6] The wheel graph 푊푛 = 퐶푛−1 ⋀ 퐾1 is a graph with 푛 vertices (푛 ≥ 4), formed by joining the central vertex 퐾1 to all the vertices of 퐶푛−1.
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Definition 2. [5] A helm graph, denoted 퐻푛 is a graph obtained by attaching a single edge and vertex to each vertex of the outer circuit of a wheel graph 푊푛. Theorem 2. The helm graph 퐻푛 admits a 2-odd labeling for all 푛 ≥ 4. Proof. Let 푊푛 be the given wheel graph on 푛 ≥ 4 vertices which consists of 푛 − 1 vertices on the rim, namely 푣1, 푣2, . . . , 푣푛 and a central vertex, say 푣0. Obtain the helm graph 퐻푛 by joining a pendent edge at each vertex of the cycle. We denote the corresponding newly added vertices as 푢1, 푢2, . . . , 푢푛. Now define a one-to-one function 푓: 푉(퐻푛) → 푍 as follows: without loss of generality, let 푓(푣0) = 1, 푓(푣1) = 3, 푓(푣2) = 4, and 푓(푣푖) = 푓(푣푖−1) + 2 for 3 ≤ 푖 ≤ 푛. Let 푘 be the first even number sufficiently larger number than 푓(푣푛). Then 푓(푢1) = 푘, 푓(푢2) = 푘 + 1, and 푓(푢푖) = 푓(푢푖−1) + 2 for 3 ≤ 푖 ≤ 푛. An easy check shows that 푓 is the required 2-odd labeling of 퐻푛 as |푓(푣0) − 푓(푣푖)| are all odd numbers for 3 ≤ 푖 ≤ 푛 , |푓(푣0) − 푓(푣1)| = 2, |푓(푣0) − 푓(푣2)| = 3 , |푓(푣푖) − 푓(푣푖+1)| = 2 for 2 ≤ 푖 ≤ 푛 − 1 , |푓(푣푛) − 푓(푣1)| is an odd integer, and |푓(푣푖) − 푓(푢푖)| are all odd integers, for 1 ≤ 푖 ≤ 푛. Corollary 1. The wheel graph 푊푛 admits a 2-odd labeling for all 푛 ≥ 4.
Figure 1. A 2-odd labeling of helm graph 퐻9
Definition 3. [9] A double-wheel graph 퐷푊푛 of size 푛 is composed of 2퐶푛⋀퐾1. In other words, 퐷푊푛 consists of two cycles, each of size 푛, where the vertices of the two cycles are all connected to a common hub represented by 퐾1. Theorem 3. The double wheel graph 퐷푊푛 admits a 2-odd labeling for 푛 ≥ 3. Proof. Let 퐷푊푛 be the given double wheel graph with 푛 ≥ 3. We label the central vertex as 푣0, the inner cycle vertices as 푣1, 푣2, . . . , 푣푛, and the outer cycle vertices as 푢1, 푢2, . . . , 푢푛. Define a one-to-one labeling ℎ: 푉(퐷푊푛) → 푍 as follows: without loss of generality, let ℎ(푣0) = 1, ℎ(푣1) = 3, ℎ(푣2) = 4, and ℎ(푣푖) = ℎ(푣푖−1) + 2 for 3 ≤ 푖 ≤ 푛. Also define ℎ(푢푖) = −ℎ(푣푖) for 1 ≤ 푖 ≤ 푛 . One can clearly observe that ℎ induces the required 2-odd labeling of 퐷푊푛 for all 푛 ≥ 3.
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Definition 4. [1]
An umbrella graph 푈(푚, 푛) is defined as a graph with 푉 (푈(푚, 푛)) = {푣1, 푣2, … , 푣푚+푛} and 퐸(푈(푚, 푛)) = 퐸1 ∪ 퐸2 ∪ 퐸3 , where 퐸1 = {푣푖푣푖+1 ∶ 1 ≤ 푖 ≤ 푛 − 1}, 퐸2 = {푣푛푣푛+푗: 1 ≤ 푗 ≤ 푚}, and 퐸3 = {푣푘푣푘+1: 푛 + 1 ≤ 푘 ≤ 푚 + 푛 − 1}. Theorem 4. An umbrella graph 푈(푚, 푛) admits a 2-odd labeling for any 푚, 푛 ∈ 푁. Proof. Let 푈(푚, 푛) be the given umbrella graph on 푚, 푛 ≥ 1 vertices. We label the vertex common vertex as 푣0 , vertices of a path 푃푚 as 푣1, 푣2, . . . , 푣푚 , and vertices of the handle as 푢1, 푢2, . . . , 푢푛 . One can note that 푣0 = 푢1 and |푈(푚, 푛)| = 푚 + 푛 + 1. Now define an injective labeling ℎ: 푉(푈(푚, 푛)) → 푍 as follows: without loss of generality, let ℎ(푣0) = 1 and ℎ(푣푖) = 2푖 for 1 ≤ 푖 ≤ 푚. Next let 2푘 be any even number sufficiently greater than ℎ(푣푚). Then define ℎ(푢2) = 2푘 and ℎ(푢푖) = 2ℎ(푢푖−1) for 3 ≤ 푖 ≤ 푛. An easy check shows that ℎ is the desired 2-odd labeling of 푈(푚, 푛) for any 푚, 푛 ∈ 푁. Definition 5: [7] ′ ′′ Duplication of a vertex 푣푘 of a graph H by an edge 푒 = 푣푘푣푘 is obtained by adding two new ′ ′′ ′ ′′ vertices 푣푘, 푣푘 and an edge 푒 = 푣푘푣푘 such that 푁(푣푘′) = {푣푘, 푣푘′′} and 푁(푣푘′′) = {푣푘, 푣푘′ }. The resultant graph is denoted by G. Conjecture 1: [2] (The Twin Prime Conjecture) There are infinitely many pairs of primes that differ by 2. Theorem 5: The graph obtained by performing duplication of a vertex by an edge at all the vertices of a 2- odd graph admits a 2-odd labeling if the Twin prime conjecture is true. Proof. Let 퐻 be the given 2-odd graph with a 2-odd labeling ℎ and 푉 (퐻) = {푣1, 푣2, . . . , 푣푛}. Let ℎ(푣푗) = max ℎ(푣푖). Obtain the graph 퐺 by performing duplication of every vertex 푣푘 by an 1≤푖≤푛 edge 푣′푘푣′′푘 for 푘 = 1, 2, . . . , 푛. One can see that G is a graph with 3n vertices and having n vertex disjoint cycles of length 3. Define a one-to-one labeling 푔 ∶ 푉 (퐺) → 푍 as follows: let 푔(푣푖) = ℎ(푣푖) for all 1 ≤ 푖 ≤ 푛. Let 푝′1 and 푝′′1 be any twin primes sufficiently larger ′ ′′ than ℎ(푣푗). Let 푔(푣 푗) = ℎ(푣푗) + 푝′1 and 푔(푣′′푗) = ℎ(푣푗) + 푝 1. Next let 푝′2, 푝′′2 be any ′ twin primes sufficiently larger than 푔(푣′′푗). Now let ℎ(푣푙) = max ℎ(푣푖). Let 푔(푣 푙) = 1≤푖≤푛,푖≠푗 ′′ ℎ(푣푙) + 푝′2 and 푔(푣′′푙) = ℎ(푣푙) + 푝 2. Continuing the same process for all the vertices of 퐺, one can obtain 2-odd labeling of 퐺 as there are infinitely many pairs of primes that differ by 2 by the Twin prime conjecture. Definition 6. [8] The join of two graphs 퐻1 and 퐻2, denoted by 퐻1 + 퐻2, is a graph formed with 푉 (퐻1 + 퐻2) = 푉(퐻1) ∪ 푉(퐻2) and 퐸(퐻1 + 퐻2) = 퐸(퐻1) ∪ 퐸(퐻2) ∪ {푢푣 ∶ 푢 ∈ 푉 (퐻1), 푣 ∈ 푉 (퐻2)}. Definition 7. [10-12] The complete graph 퐾푛 is a graph in which any two vertices are adjacent. Theorem 6. The graph 퐾2 + 푚퐾1 admits a 2-odd labeling for all 푚 ∈ 푁. Proof. Let 퐾1 and 퐾2 be the given complete graphs on 1 and 2 vertices, respectively. We label the vertices of 퐾2 as 푣1 and 푣2. We also label the vertices of 푚퐾1 as 푢1, 푢2, … , 푢푚. Obtain 퐾2 + 푚퐾1 by using Definition 6. Now define a one-one function ℎ: 푉 (퐾2 + 푚퐾1) → 푍 as follows: without loss of generality, let ℎ(푣1) = 1 and ℎ(푣2) = −1. Also define ℎ(푢푖) =
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2푖 for 1 ≤ 푖 ≤ 푚. One can see that ℎ is the required 2-odd labeling of 퐾2 + 푚퐾1 for any 푚 ∈ 푁. Conjecture 2. [2] (Goldbach’s Conjecture) Any even number 2푛 for 푛 ≥ 2 is the sum of two primes. Definition 8. [4] A graph 퐵푛,푚 (푛, 푚 ∈ 푁) is said to be a butterfly graph if two cycles of same order 푛 ≥ 3 sharing a common vertex with an arbitrary number 푚 of pendant edges attached at the common vertex. Theorem 7. A butterfly graph 퐵푛,푚 permits a 2-odd labeling for any 푛 ≥ 3, 푚 ∈ 푁 if Goldbach’s conjecture is true. Proof. Let 퐵푛,푚 be the given butterfly graph with 푛 ≥ 3 and 푚 ∈ 푁. We call the vertex set of 퐵푛,푚 as 푉 (퐵푛,푚) = {푣1, 푣2, … , 푣푛} ∪ {푢1, 푢2, … , 푢푛} ∪ {푤1, 푤2, … , 푤푚} . Clearly |푉 (퐵푛,푚)| = 2푛 − 1 + 푚 as 푢1 = 푣1. We consider the following two cases. Case 1: 퐶푛, when 푛 is even The proof is direct from Theorem 1. Case 2: 퐶푛, when 푛 is odd Define an injective labeling ℎ: 푉 (퐵푛,푚) → 푍 as follows: without loss of generality, let ℎ(푣푖) = 2(푖 − 1) for 1 ≤ 푖 ≤ 푛 − 1. Next by Goldbach’s conjecture, ℎ(푣푛−1) = 푝1 + 푝2 and let ℎ(푣푛) = 푝1 or 푝2 . Also define ℎ(푢푖) = − ℎ(푣푖) for all 2 ≤ 푖 ≤ 푛. Finally, let 푙1, 푙2, … , 푙푚 be sufficiently larger odd numbers than ℎ(푣푛−1) and ℎ(푤푖) = 푙푖 for all 1 ≤ 푖 ≤ 푚. A simple check shows that the vertex labels are distinct and ℎ induces a 2-odd labeling of 퐵푛,푚. Definition 9. [7] Duplication of a vertex 푣 of a graph 퐻 by a vertex is obtained by adding a new vertex 푤 to 퐻 and adding edges in such a way that 푁(푤) = 푁(푣). The resultant graph is denoted by 퐺. Theorem 8. The graph obtained from a cycle 퐶푛 by performing duplication of a vertex by a vertex at all the vertices of 퐶푛 admits a 2-odd labeling if Goldbach’s conjecture is true. Proof. Let 퐶푛: 푣1푣2, … , 푣푛푣1 be the given cycle. Obtain the graph 퐺 by performing duplication of a vertex by a vertex at all the vertices of 퐶푛 with 푉 (퐺) = 푉 (퐶푛) ∪ {푢푖: 1 ≤ 푖 ≤ 푛}. Define an injective function ℎ ∶ 푉 (퐺) → 푍 as follows: ℎ(푣푖) = 2(푖 − 1) for 1 ≤ 푖 ≤ 푛 − 1. If Goldbach’s conjecture is true then ℎ(푣푛−1) can be expressed as a sum of two primes. Let ℎ(푣푛−1) = 푝1 + 푝2 and without loss of generality, let ℎ(푣푛) = 푝1 > 2 and ℎ(푢푛) = 푝2. Next let ℎ(푢1) = ℎ(푢1) − 2, ℎ(푢1) = ℎ(푢푛−1) + 2, and ℎ(푢푖) = 2푚 + 1 for 2 ≤ 푖 ≤ 푛 − 2 and any sufficiently large 푚 ∈ 푁. One can easily see that ℎ is the desired 2-odd labeling of 퐺. 3. Conclusion The 2-odd labeling of various classes of graphs such as helm graph, umbrella graph are established. Investigating 2-odd labeling of other classes of graphs is still open and this is for future research. One can also explore the exclusive applications of 2-odd labeling in real life situations. References [1] G. Amuda and S. Meena, Cube Difference Labeling Of Some Cycle-related Graphs, International Journal of Innovative Science, Engineering & Technology, 2(1) (2015), 461- 471.
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[2] Kenneth H. Rosen, Elementary Number Theory and its Applications, sixth ed., Addison- Wesley, Reading, MA, 2010. [3] J. D. Laison, C. Starr, and A. Walker, “Finite Prime Distance Graphs and 2-odd Graphs”, Discrete Mathematics, vol. 313, pp. 2281-2291, 2013. [4] S. Meena and P. Kavitha, Prime Labeling for Some Butterfly Related Graphs, International Journal of Mathematical Archive, 5(10), pp. 15-25, 2014. [5] S. Meena and K. Vaithilingam, Prime Labeling for Some Helm Related Graphs, International Journal of Innovative Research in Science, Engineering and Technology, 2(4), (2013), 1075-1085. [6] A. Parthiban and N. Gnanamalar David, “On Finite Prime Distance Graphs”, Indian Journal of Pure and Applied Mathematics, 2017. (To appear) [7] A. Parthiban and N. Gnanamalar David, “Prime Distance Labeling of Some Path Related graphs”, International Journal of Pure and Applied Mathematics, vol. 120 (7), pp. 59-67, 2018. [8] R. Ponraj, S. Sathish Narayanan, and R. Kala, A Note on Difference Cordial Graphs, Palestine Journal of Mathematics, 4(1), pp. 189-197, 2015. [9] Ronan Le Bras, Carla P. Gomes, and Bart Selman, Double-Wheel Graphs Are Graceful, Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence, (2013), 587-593. [10] D. B. West, Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2000. [11] Singh, G., Gupta, M. K., Mia, M., & Sharma, V. S. (2018). Modeling and optimization of tool wear in MQL-assisted milling of Inconel 718 superalloy using evolutionary techniques. The International Journal of Advanced Manufacturing Technology, 97(1-4), 481- 494. [12] Kant, N., Wani, M. A., & Kumar, A. (2012). Self-focusing of Hermite–Gaussian laser beams in plasma under plasma density ramp. Optics Communications, 285(21-22), 4483- 4487.
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