In a graph G a mapping g is known as odd prime labeling , if g is abijection from V to f1; 3; 5; ::::; 2jVj - 1g satisfying the condition thatfor each line xy in G the gcd of the labels of end points (g(x); g(y)) isone. In this article we prove that some new arrow related graphs suchas A2y, A3 y,A5 y, are all odd prime graphs. Also we prove that doublearrow graphs, DA2 y and DA3 y are odd prime graphs

G, Gajalakshmi

S, Meena

English

APAV - Academy of Sciences, Letters, Arts and Technology (E-Journals)

Ratio Mathematica Volume 46, 2023Odd prime labeling for some arrow relatedgraphsG. Gajalakshmi*S. Meena†AbstractIn a graph G a mapping g is known as odd prime labeling , if g is abijection from V to {1, 3, 5, ...., 2|V| − 1} satisfying the condition thatfor each line xy in G the gcd of the labels of end points (g(x), g(y)) isone. In this article we prove that some new arrow related graphs suchas A2y, A3y,A5y, are all odd prime graphs. Also we prove that doublearrow graphs, DA2y and DA3y are odd prime graphs.Keywords: Prime graph, Odd prime graph, Arrow graphs.2020 AMS subject classifications: 05C78 1*Department of Mathematics, Govt, Arts & Science College, Chidambaram;gaja61904@gmail.com.†Department of Mathematics, Govt, Arts & Science College, Chidambaram 608 102, India;meenasaravanan14@gmail.com.1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20,2023. DOI: 10.23755/rm.v46i0.1059. ISSN: 1592-7415. eISSN: 2282-8214. ©G. Gajalakshmiet al. This paper is published under the CC-BY licence agreement.70G. Gajalakshmi and S. Meena1 IntroductionIn this article by a graph G = 〈V (G), E(G)〉 we mean a simple graph. Forgraph theoretical notations we refer J.A.Bondy and U .S. R.Murthy [1976] .Graph labeling has been introduced in mid 1960. For entire survey of graphlabeling we refer Gallian [2015].The concept of prime labeling was established by Roger Entringer and wasdiscussed in a article by Deretsky et al. [1991], Tout et al. [1982]. A graph G oforder p is known as prime graph if it’s points can be labeled with distinct positiveintegers {1, 2, 3, ·, p} such that the labels of any two adjacent points are relativelyprime Meena and Vaithilingam [2013]. Meena and Kavitha [2014] investigatedprime labeling for some butterfly related graphs. Meena et al. [2021] investigatedodd prime labeling for some new classes of graph.The notion of odd prime labeling was established by Prajapati and Shah [2018]and many researchers. Arrow graph was introduced by Kaneria et al. [2015]. Mo-tivated by this study, in this article investigate the existence of odd prime labelingof some graphs related to arrow graphs.Definition 1.1. LetH = 〈V(H), E(H)〉 be a graph. A bijection g : V(H)→ O|V |is know as odd prime labeling if for each line xy ∈ E , greatest common divisor〈g(x), g(y)〉 = 1. A graph is know as odd prime graph if its admits odd primelabeling .Definition 1.2. LetH1 = (P1, Q1) andH2 = (P2, Q2) be two graphs withP1 ∩ P2 = φ. The cartesian product H1 × H2 is defined as a graph havingP = P1 × P2 and x = (x1, x2) and y = (y1, y2) are adjacent if x1 = y1 and x2is adjacent to y2 in H2 or x1 is adjacent to y1 in H1 and x2 = y2. The cartesianproduct of two paths Pm and Pn denoted as Pm×Pn is known as a grid graph onnm points and 2nm− (n+m).Definition 1.3. In rectangular grid Pm × Pn on mn points the n pointsv1,1, v2,1, v3,1....vm,n and points v1,n, v2,n, v3,n....vm,n are called an superior pointsfrom both the ends.Definition 1.4. An arrow graphAxy with width x and length y is got by connectinga point v with superior points of Px × Py by new edges from one end.Definition 1.5. A double arrow graph DAxy with width x and length y is got byconecting two points v and w with superior points of Pm×Py by x+x new edgesfrom both the end.71Odd prime labeling for some arrow related graphs2 Main ResultsTheorem 2.1. A2y is an odd prime graph where y ≥ 2.Proof. Let G = A2y be an arrow graph got by connecting a point g(u0) withsuperior points of P2 × Py by new lines.Let V(G) = {ul/0 ≤ l ≤ y} ∪ {vl/1 ≤ l ≤ y}E(G) = {ulul+1/1 ≤ l ≤ y − 1} ∪ {u0v1} ∪ {u0u1}∪ {vlvl+1/1 ≤ l ≤ y − 1} ∪ {ulvl/1 ≤ l ≤ y}.Now |V(G)| = 2y+1 and |E(G)| = 3yDefine a Mapping f : V → O2y as followsg(u0) = 1g(ul) = 4l − 1 for 1 ≤ l ≤ yg(vl) = 4l + 1 for 1 ≤ l ≤ yClearly point labels are distinct.For each e ∈ E, if gcd(g(u), g(v)) = 1(i) e = u0ul, gcd(g(u0), g(ul)) = gcd(1, 3) = 1(ii) e = u0v1, gcd(g(u0), g(vl)) = gcd(1, 5) = 1(iii)e = ulvl, gcd(g(ul), g(vl)) = gcd(4l − 1, 4l + 1) = 1for 1 ≤ l ≤ y(iv) e = ulul+1, gcd(g(ul), g(ul+1))= gcd(4l − 1, 4l + 3) = 1for 1 ≤ l ≤ y − 1(v) e = vivl+1, gcd(g(vl)), g(vl+1) = gcd(4l + 1, 4l + 5) = 1for 1 ≤ l ≤ y − 1Hence A2y is an odd prime graph.Figure 1: Arrow graph A2y and its odd prime labelingTheorem 2.2. A3y is an odd prime graph where y ≥ 2.Proof. Let G = A3y be an arrow graph got by connecting a point g(u0) withsuperior points of P3 × P2 by 3 new lines.V(G) = {ul, vl, wl, /1 ≤ l ≤ y} ∪ {u0}E(G) = {ulul+1, vlvl+1, wlwl+1/1 ≤ l ≤ y − 1}∪{vlwl, ulvl/1 ≤ l ≤ y}∪{uou1}∪{u0v1}∪{u0w1}72G. Gajalakshmi and S. MeenaNow |V(G)| = 3y + 1 and |E(G)| = 5y − 1Define a mapping f : V → O2y as followsg(u0) = 1g(ul) = 6l − 3 for 1 ≤ l ≤ y, l is oddg(ul) = 6l − 1 for 1 ≤ l ≤ y, l is eveng(vl) = 6l − 1 for 1 ≤ l ≤ y, l is oddg(vl) = 6l − 3 for 1 ≤ l ≤ y, l is eveng(wl) = 6l + 1 for 1 ≤ l ≤ yClearly all the point labels are distinct. With this labeling for each e = uv ∈ E if(i) e = u0u1, gcd(g(u0), g(u1)) = gcd(1, 3) = 1 for 1 ≤ l ≤ y(ii) e = u0w1, gcd(g(u0), g(w1)) = gcd(1, 7) = 1 for 1 ≤ l ≤ y(iii)e = u0v1, gcd(g(u0), g(v1)) = gcd(1, 5) = 1 for 1 ≤ l ≤ y(iv)e = ulvl, gcd(g(ul), g(vl) = gcd(6l − 3, 6l − 1) = 1 for 1 ≤ l ≤ yl 6≡ 0(mod2)(v)e = ulvl, gcd(g(ul), g(vl)) = gcd(6l − 3, 6l − 1) = 1 for 1 ≤ l ≤ yl ≡ 0(mod2)(vi)e = vlwl, gcd(g(vl), g(wl)) = gcd(6l − 1, 6l + 1) = 1 for 1 ≤ l ≤ yl 6≡ 0(mod2)(vii)e = vlwl, gcd(g(ul), g(vl)) = gcd(6l − 1, 6l − 3) = 1 for 1 ≤ l ≤ yl ≡ 0(mod2)(viii)e = ulul+1, gcd(g(ul), g(ul+1)) = gcd(6l − 3, 6l − 5) = 1 for 1 ≤ l ≤ yl 6≡ 0(mod2)(ix)e = ulul+1, gcd(g(ul), g(ul+1)) = gcd(6l − 1, 6l + 3) = 1 for 1 ≤ l ≤ yl ≡ 0(mod2)(x)e = vlvl + 1, gcd(g(vl), g(vl+1)) = gcd(6l − 3, 6l − 1) = 1 for 1 ≤ l ≤ yl ≡ 0(mod2)(xi) e = vlvl + 1, gcd(g(vl), g(vl+1)) = gcd(6l − 1, 6l − 3) = 1 for 1 ≤ l ≤ yl 6≡ 0(mod2)(xii) e = wlwl + 1, gcd(g(wl), g(wl+1)) = gcd(6l + 1, 6l + 7) = 1 for 1 ≤ l ≤ yHence A3y is an odd prime graph .Figure 2: Arrow graph A3y and its odd prime labelingTheorem 2.3. A5y is an odd prime graph where y ≥ 5.73Odd prime labeling for some arrow related graphsProof. Let G = A5y be an arrow graph got by connecting a point v with superiorpoints P5 × Py by 5 new lines.V(G) = {ul, vl, wl/1 ≤ l ≤ y} ∪{u0}E(G) = {ulvl, vlwl/1 ≤ l ≤ y}∪{(ulul+1), (vlvi+1), (wlwl+1/1 ≤ l ≤ y − 1}Now |V(G)| = 5y + 1 and |E(G)| = 9yDefine a mapping f : V → Oy as followsg(u0) = 1g(ul) = 6l − 3 for 1 ≤ l ≤ l, l is oddg(ul) = 6l − 1 for 1 ≤ l ≤ l, l is eveng(vl) = 6l − 1 for 1 ≤ l ≤ l, l is oddg(vl) = 6l − 3 for 1 ≤ l ≤ l, l is eveng(wl) = 6l + 1 for 1 ≤ l ≤ l,Clearly all the point labels are distinct. With this labeling for each e ∈ E ifgcd(g(u), g(v)) = 1(i) e = u0u1, gcd(g(u0), g(u1)) = gcd(1, 3) = 1(ii) e = u0ul+1, gcd(g(u0), g(ul+1)) = gcd(1, 6l − 3) = 1 for 1 ≤ l ≤ y(iii)e = ulvl, gcd(g(ul), g(vl) = gcd(6l − 3, 6l − 1) = 1 for 1 ≤ l ≤ yl 6≡ 0(mod2)(iv) e = ulvl, gcd(g(ul), g(vl)) = gcd(6l − 1, 6l − 3) = 1 for 1 ≤ l ≤ yl ≡ 0(mod2)(v) e = vlwl, gcd(g(vl), g(wl)) = gcd(6l − 1, 6l + 1) = 1 for 1 ≤ l ≤ yl 6≡ 0(mod2)(vi) e = vlwl, gcd(g(vl), g(wl)) = gcd(6l − 3, 6l + 1) = 1 for 1 ≤ l ≤ yl ≡ 0(mod2)(vii) e = ulul+1, gcd(g(ul), g(ul+1)) = gcd(6l − 3, 6l + 5) = 1 for 1 ≤ l ≤ yl 6≡ 0(mod2)(viii) e = ulul+1, gcd(g(ul), g(ul+1)) = gcd(6l − 1, 6l − 3) = 1 for 1 ≤ l ≤ yl ≡ 0(mod2)(ix) e = vlvl+1, gcd(g(vl), g(vl+1)) = gcd(6l − 1, 6l + 3) = 1 for 1 ≤ l ≤ yl 6≡ 0(mod2)(x) e = vivi+1, gcd(g(vl), g(vl+1)) = gcd(6l − 3, 6l + 5) = 1 for 1 ≤ l ≤ yl ≡ 0(mod2)(xi) e = wlwl+1, gcd(g(wl), g(wl+1)) = gcd(6l + 1, 6l + 7) = 1 for 1 ≤ l ≤ yHence A5y is an odd prime graph .74G. Gajalakshmi and S. MeenaFigure 3: Arrow graph A5y and its odd prime labelingTheorem 2.4. DA2y is an odd prime graph where y ≥ 2.Proof. Let G = DA2y be a double arrow graph got by connecting two points u, vwith superior points from both the ends of P2 × Py by 2+2 new lines.Let V(G) = {ulvl/1 ≤ l ≤ y} ∪ {v, v0}E(G) = {(ulul+1), (vlvl+1), 1 ≤ l ≤ y − 1} ∪ {vlul/1 ≤ l ≤ y} ∪{vv1}∪{vu1}∪{uyv0}∪{vyv0}Now |V(G)| = 2y+2 and |E(G)| = 3y+4Define a mapping f : V → O2y as followsg(v) = 1g(ui) = 4l − 1 for 1 ≤ l ≤ yg(vi) = 4l + 1 for 1 ≤ l ≤ yg(v0)= 4y + 3Clearly point labels are distinct.For every e = uv ∈ E, if gcd(g(u), g(v)) = 1(i) e = vu1, gcd(g(v), g(u1)) = gcd(1, 3) = 1(ii) e = vv1, gcd(g(v), g(v1)) = gcd(1, 5) = 1(iii) e = ulul+1, gcd(g(ul), g(ul+1))= gcd(4l − 1, 4l + 3) = 1 for 1 ≤ l ≤ y − 1(iv) e = vlvl+1, gcd(g(vl)), g(vl+1) = gcd(4l + 1, 4l + 5) = 1 for 1 ≤ l ≤ y − 1(v) e = vlul, gcd(g(vl), g(ul)) = gcd(4l + 1, 4l − 1) = 1 for 1 ≤ l ≤ y(vi) e = vyw, gcd(g(vy), g(w)) = gcd(4y + 1, 4y + 3) = 1(vii)e = uyw, gcd(g(uy), g(w)) = gcd(4y − 1, 4y + 3) = 1Hence DA2y is an odd prime graph.75Odd prime labeling for some arrow related graphsFigure 4: Arrow graph DA2y and its odd prime labelingTheorem 2.5. DA3y is an odd prime graph where y ≥ 3.Proof. Let D = DA3y be an arrow graph got by connecting two point set u0 and z0with superior pointss from both the ends of P3 × P2 by 3+3 new lines.V(G) = {ul, vl, wl, /1 ≤ l ≤ y} ∪ {u0} ∪ {z0}E(G) = {ulul+1, vlvl+1, wlwl+1/1 ≤ l ≤ y − 1}∪ {wlvl, vlul/1 ≤ l ≤ y}∪{u0u1, u0v1, u0w1, z0uy, z0vy, z0wy}Now |V(G)| = 3y + 2 and |E(G)| = 5y + 3Define a mapping f : V → Oy as followsg(u0) = 1g(ul) = 6l − 3 for 1 ≤ l ≤ y, l is oddg(ul) = 6l − 1 for 1 ≤ l ≤ y, l is eveng(vl) = 6l − 1 for 1 ≤ l ≤ y, l is oddg(vl) = 6l − 3 for 1 ≤ l ≤ y, l is eveng(wl) = 6l + 1 for 1 ≤ l ≤ yg(z0) = 6y + 3 for 1 ≤ i ≤ yClearly all the point values are different. With this labeling for each e ∈ E if(i) e = u0u1, gcd(g(u0), g(u1)) = gcd(1, 3) = 1(ii) e = u0v1, gcd(g(u0), g(v1)) = gcd(1, 5) = 1(iii) e = u0w1, gcd(g(u0), g(w1) = gcd(1, 7) = 1(iv) e = ulvl, gcd(g(ul), g(vl)) = gcd(6l − 3, 6l − 1) = 1 for 1 ≤ l ≤ yl 6≡ 0(mod2)(v) e = ulvl, gcd(g(ul), g(vl)) = gcd(6l − 1, 6l − 3) = 1 for 1 ≤ l ≤ y l ≡0(mod2);(vi) e = vlwl, gcd(g(vl), g(wl)) = gcd(6l − 1, 6l + 1) = 1 for 1 ≤ l ≤ yl 6≡ 0(mod2)(vii) e = vlwl, gcd(g(vl), g(wl)) = gcd(6l − 3, 6l − 1) = 1 for 1 ≤ l ≤ yl ≡ 0(mod2);(viii) e = ulul+1, gcd(g(ul), g(ul+1)) = gcd(6l− 3, 6l+5) = 1 for 1 ≤ l ≤ y− 1l 6≡ 0(mod2);(ix) e = ulul+1, gcd(g(ul), g(ul+1)) = gcd(6l − 1, 6l + 3) = 1 for 1 ≤ l ≤ y − 1l ≡ 0(mod2);Hence DA3y is an odd prime graph .76G. Gajalakshmi and S. MeenaFigure 5: Arrow graph DA3n and its odd prime labeling3 ConclusionsThe odd Prime labeling of various classes of graphs such as A2y where y ∈ N,A3y, A5y, where y ≥ 2 are odd prime graph and double arrow graphs DA2y,DA3yare proved. To derive similar results for other graph families is an open area ofresearch.ReferencesT. Deretsky, S. Lee, and J. Mitchem. On vertex prime labelings of graphs, ingraph theory, combinatorics and applications, j. alavi, g. chartrand, o. oeller-man, and a. schwenk, eds.,. Proceedings 6th International Conference Theoryand Applications of Graphs (Wiley, New York),, 1:359 – 369, 1991.J. Gallian. A dynamic survey of graph labeling. DS6, 2015.J.A.Bondy and U .S. R.Murthy. Graph Theory and Applications. (North-Holland),New York, 1976.V. Kaneria, M. Jariya, and H. Makadia. Graceful of arrow graphs and doublearrow graph. Malaya Journal of Math., 3:382 – 386, 2015.S. Meena and P. Kavitha. Prime labeling for some butterfly related graphs. Inter-national Journal of Mathematical Archive, 5:15 – 25, 2014.S. Meena and K. Vaithilingam. Prime labeling for some crown related graphs.International Journal of Scientific & Technology Research, 2:92 – 95, 2013.S. Meena, G. Gajalakshmi, and P. Kavitha. Odd prime labeling for some newclasses of graph (communicated). SEAJM, 2021.77Odd prime labeling for some arrow related graphsU. Prajapati and K. Shah. On odd prime labeling. International journal of Re-search and Analytical Reviews, 5:284 – 294, 2018.A. Tout, A. Dabboucy, and K. Howalla. Prime labeling of graphs. Nat. Acad. Sciletters, 11:365 – 368, 1982.78

Odd Prime Labeling For Some Arrow Related Graphs

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Odd Prime Labeling For Some Arrow Related Graphs

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