Ratio Mathematica Volume 46, 2023

Odd prime labeling for some arrow related
graphs

G. Gajalakshmi*
S. Meena†

Abstract

In a graph G a mapping g is known as odd prime labeling , if g is a
bijection from V to {1,3,5, ....,2|V|−1} satisfying the condition that
for each line xy in G the gcd of the labels of end points (g(x),g(y)) is
one. In this article we prove that some new arrow related graphs such
as A2y, A

3
y,A

5
y, are all odd prime graphs. Also we prove that double

arrow 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. Gajalakshmi
et al. This paper is published under the CC-BY licence agreement.

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G. Gajalakshmi and S. Meena

1 Introduction
In this article by a graph G = 〈V (G),E(G)〉 we mean a simple graph. For

graph 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 graph

labeling we refer Gallian [2015].
The concept of prime labeling was established by Roger Entringer and was

discussed in a article by Deretsky et al. [1991], Tout et al. [1982]. A graph G of
order p is known as prime graph if it’s points can be labeled with distinct positive
integers {1,2,3, ·,p} such that the labels of any two adjacent points are relatively
prime Meena and Vaithilingam [2013]. Meena and Kavitha [2014] investigated
prime labeling for some butterfly related graphs. Meena et al. [2021] investigated
odd 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 labeling
of some graphs related to arrow graphs.

Definition 1.1. Let H = 〈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 prime
labeling .

Definition 1.2. Let H1 = (P1,Q1) and H2 = (P2,Q2) be two graphs with
P1 ∩ P2 = φ. The cartesian product H1 × H2 is defined as a graph having
P = P1 × P2 and x = (x1,x2) and y = (y1,y2) are adjacent if x1 = y1 and x2
is adjacent to y2 in H2 or x1 is adjacent to y1 in H1 and x2 = y2. The cartesian
product of two paths Pm and Pn denoted as Pm ×Pn is known as a grid graph on
nm points and 2nm− (n + m).

Definition 1.3. In rectangular grid Pm ×Pn on mn points the n points
v1,1,v2,1,v3,1....vm,n and points v1,n,v2,n,v3,n....vm,n are called an superior points
from both the ends.

Definition 1.4. An arrow graph Axy with width x and length y is got by connecting
a 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 by
conecting two points v and w with superior points of Pm×Py by x+x new edges
from both the end.

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Odd prime labeling for some arrow related graphs

2 Main Results
Theorem 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) with
superior 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)| = 3y
Define a Mapping f : V → O2y as follows
g(u0) = 1
g(ul) = 4l−1 for 1 ≤ l ≤ y
g(vl) = 4l + 1 for 1 ≤ l ≤ y
Clearly 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) = 1

for 1 ≤ l ≤ y
(iv) e = ulul+1, gcd(g(ul),g(ul+1))= gcd(4l−1,4l + 3) = 1
for 1 ≤ l ≤ y −1
(v) e = vivl+1, gcd(g(vl)),g(vl+1) = gcd(4l + 1,4l + 5) = 1
for 1 ≤ l ≤ y −1
Hence A2y is an odd prime graph.

Figure 1: Arrow graph A2y and its odd prime labeling

Theorem 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) with
superior 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}

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Now |V(G)| = 3y + 1 and |E(G)| = 5y −1
Define a mapping f : V → O2y as follows
g(u0) = 1
g(ul) = 6l−3 for 1 ≤ l ≤ y, l is odd
g(ul) = 6l−1 for 1 ≤ l ≤ y, l is even
g(vl) = 6l−1 for 1 ≤ l ≤ y, l is odd
g(vl) = 6l−3 for 1 ≤ l ≤ y, l is even
g(wl) = 6l + 1 for 1 ≤ l ≤ y
Clearly 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 ≤ y
l 6≡ 0(mod2)
(v)e = ulvl,gcd(g(ul),g(vl)) = gcd(6l − 3,6l − 1) = 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 ≤ y
l 6≡ 0(mod2)
(vii)e = vlwl,gcd(g(ul),g(vl)) = gcd(6l − 1,6l − 3) = 1 for 1 ≤ l ≤ y
l ≡ 0(mod2)
(viii)e = ulul+1,gcd(g(ul),g(ul+1)) = gcd(6l−3,6l−5) = 1 for 1 ≤ l ≤ y
l 6≡ 0(mod2)
(ix)e = ulul+1,gcd(g(ul),g(ul+1)) = gcd(6l−1,6l + 3) = 1 for 1 ≤ l ≤ y
l ≡ 0(mod2)
(x)e = vlvl + 1,gcd(g(vl),g(vl+1)) = gcd(6l−3,6l−1) = 1 for 1 ≤ l ≤ y
l ≡ 0(mod2)
(xi) e = vlvl + 1,gcd(g(vl),g(vl+1)) = gcd(6l−1,6l−3) = 1 for 1 ≤ l ≤ y
l 6≡ 0(mod2)
(xii) e = wlwl + 1,gcd(g(wl),g(wl+1)) = gcd(6l + 1,6l + 7) = 1 for 1 ≤ l ≤ y
Hence A3y is an odd prime graph .

Figure 2: Arrow graph A3y and its odd prime labeling

Theorem 2.3. A5y is an odd prime graph where y ≥ 5.

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Odd prime labeling for some arrow related graphs

Proof. Let G = A5y be an arrow graph got by connecting a point v with superior
points 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)| = 9y
Define a mapping f : V → Oy as follows
g(u0) = 1
g(ul) = 6l−3 for 1 ≤ l ≤ l, l is odd
g(ul) = 6l−1 for 1 ≤ l ≤ l, l is even
g(vl) = 6l−1 for 1 ≤ l ≤ l, l is odd
g(vl) = 6l−3 for 1 ≤ l ≤ l, l is even
g(wl) = 6l + 1 for 1 ≤ l ≤ l,
Clearly all the point labels are distinct. With this labeling for each e ∈ E if
gcd(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 ≤ y
l 6≡ 0(mod2)
(iv) e = ulvl,gcd(g(ul),g(vl)) = gcd(6l − 1,6l − 3) = 1 for 1 ≤ l ≤ y
l ≡ 0(mod2)
(v) e = vlwl,gcd(g(vl),g(wl)) = gcd(6l − 1,6l + 1) = 1 for 1 ≤ l ≤ y
l 6≡ 0(mod2)
(vi) e = vlwl,gcd(g(vl),g(wl)) = gcd(6l − 3,6l + 1) = 1 for 1 ≤ l ≤ y
l ≡ 0(mod2)
(vii) e = ulul+1,gcd(g(ul),g(ul+1)) = gcd(6l − 3,6l + 5) = 1 for 1 ≤ l ≤ y
l 6≡ 0(mod2)
(viii) e = ulul+1,gcd(g(ul),g(ul+1)) = gcd(6l−1,6l−3) = 1 for 1 ≤ l ≤ y
l ≡ 0(mod2)
(ix) e = vlvl+1,gcd(g(vl),g(vl+1)) = gcd(6l − 1,6l + 3) = 1 for 1 ≤ l ≤ y
l 6≡ 0(mod2)
(x) e = vivi+1,gcd(g(vl),g(vl+1)) = gcd(6l − 3,6l + 5) = 1 for 1 ≤ l ≤ y
l ≡ 0(mod2)
(xi) e = wlwl+1,gcd(g(wl),g(wl+1)) = gcd(6l + 1,6l + 7) = 1 for 1 ≤ l ≤ y
Hence A5y is an odd prime graph .

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G. Gajalakshmi and S. Meena

Figure 3: Arrow graph A5y and its odd prime labeling

Theorem 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,v
with 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+4
Define a mapping f : V → O2y as follows
g(v) = 1
g(ui) = 4l−1 for 1 ≤ l ≤ y
g(vi) = 4l + 1 for 1 ≤ l ≤ y
g(v0)= 4y + 3
Clearly 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) = 1
Hence DA2y is an odd prime graph.

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Odd prime labeling for some arrow related graphs

Figure 4: Arrow graph DA2y and its odd prime labeling

Theorem 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 z0
with 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 + 3
Define a mapping f : V → Oy as follows
g(u0) = 1
g(ul) = 6l−3 for 1 ≤ l ≤ y, l is odd
g(ul) = 6l−1 for 1 ≤ l ≤ y, l is even
g(vl) = 6l−1 for 1 ≤ l ≤ y, l is odd
g(vl) = 6l−3 for 1 ≤ l ≤ y, l is even
g(wl) = 6l + 1 for 1 ≤ l ≤ y
g(z0) = 6y + 3 for 1 ≤ i ≤ y
Clearly 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 ≤ y
l 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 ≤ y
l 6≡ 0(mod2)
(vii) e = vlwl,gcd(g(vl),g(wl)) = gcd(6l − 3,6l − 1) = 1 for 1 ≤ l ≤ y
l ≡ 0(mod2);
(viii) e = ulul+1,gcd(g(ul),g(ul+1)) = gcd(6l−3,6l + 5) = 1 for 1 ≤ l ≤ y−1
l 6≡ 0(mod2);
(ix) e = ulul+1,gcd(g(ul),g(ul+1)) = gcd(6l − 1,6l + 3) = 1 for 1 ≤ l ≤ y − 1
l ≡ 0(mod2);
Hence DA3y is an odd prime graph .

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G. Gajalakshmi and S. Meena

Figure 5: Arrow graph DA3n and its odd prime labeling

3 Conclusions
The odd Prime labeling of various classes of graphs such as A2y where y ∈ N,

A3y, A
5
y, where y ≥ 2 are odd prime graph and double arrow graphs DA2y,DA3y

are proved. To derive similar results for other graph families is an open area of
research.

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Odd prime labeling for some arrow related graphs

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