=Paper=
{{Paper
|id=Vol-3244/PAPER_03
|storemode=property
|title=Some New Cordial Digraphs
|pdfUrl=https://ceur-ws.org/Vol-3244/PAPER_03.pdf
|volume=Vol-3244
|authors=Sarang Sadawarte,Sweta Srivastav,Rajiv Kumar
}}
==Some New Cordial Digraphs==
https://ceur-ws.org/Vol-3244/PAPER_03.pdf
Some New Cordial Digraphs
Sarang Sadawarte a, Sweta Srivastav a and Rajiv Kumar b
a
Department of Mathematics, Sharda University, Greater Noida, India
b
G L Bajaj Institute of Technology and Management, Greater Noida, India
Abstract
A directed graph, also known as a digraph, is a graph in which each edge has direction. A
linear directed cycle, Cm is a cycle whose all edges have the same direction. A linear directed
graph is called directed cordial if it preserves binary or 0 – 1 labeling under certain condition.
This paper dealt with directed cordial labeling of directed cycle Cm and square of directed
Cm. We investigate directed path joining two copies of directed cycle Cm and directed
square cycle Cm2 is directed cordial. Further we show that the directed path union of of r-
copies of directed cycle and directed square cycle graph Cm2 is cordial under certain
condition.
Keywords 1
Directed cycle, directed square cycle, cordial graph, directed cordial graph
1. Introduction
Graph theory is an important domain of discrete mathematics with voluminous applications in various
streams. Graph labeling is an allotment of labels to edges or vertices or both. Numerous graph
labeling schemes are studied and researched by many researchers. Gallian [3] reviewed and surveyed
various labeling schemes invented by many researchers. In graph theory cordial labeling plays a vital
role. It has ample of applications in many streams such as computer science, networking and
communication network. Cahit [1] introduced cordial labeling in 1958. The directed cordial labeling
of directed path was investigated and studied by Al-Shamiri [5] in 2019. He proved many results on
linear directed path in the context of some graph operations are directed cordial. We study directed
cordial labeling of directed cycle and their square subject to certain conditions. An overview provided
on basic terminology and notations is needed for the presentation of our results. In the present work,
we consider the W is directed graph.
2. Terminology and Notation
2.1. Definition
A Graph W is said to be linearly directed if all the edges have same direction. (clockwise or
anticlockwise).
2.2. Definition
Consider W as digraph. Define a function 𝜌: V (W) → {0, 1}
We define the edge labeling as follows 𝜌* : E(W) → {0, 1}
𝜌*(si si+1) =2 si (mod 2)
WCNC-2022: Workshop on Computer Networks and Communications, April 22 – 24, 2022, Chennai, India.
EMAIL: 2020489011.sadawarte@dr.sharda.ac.in (Sarang Sadawarte)
ORCID: 0000-0002-4223-6949 (Sarang Sadawarte)
©️ 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
28
𝑠𝜌 (0) and 𝑠𝜌 (1) represent number of vertices are assigned with label 0 and label 1 respectively.
Similarly, 𝑒𝜌 (0) and 𝑒𝜌 (1) represent number of edges are assigned with label 0 and label 1
respectively. This binary labeling is known as cordial if both criteria preserves
a) |𝑠𝜌 (0) −𝑠𝜌 (1)| ≤ 1
b) |𝑒𝜌 (0) −𝑒𝜌 (1)| ≤ 1
The graph W which preserves cordiality is called as a cordial graph [1].
If directed graph is cordial, we call it as directed cordial graph.
2.3. Definition
A path Pr = b0, b1….br-1 is an alternating sequence of different vertices with r – 1 length.
The linear directed path Pr have the same orientation. (clockwise or anticlockwise).
2.4. Definition
A cycle Cm is a closed path. A cycle is said to be linear directed cycle Cm if all its edges have the
same direction. (clockwise or anticlockwise).
2.5. Definition
A linear directed cycle graph Cm on m vertices is called directed square cycle graph if each pair of
vertex has distance less than or equal to two. We denote directed square cycle graph as Cm2 .
2.6. Definition
The graph W = {2-Cm : Pr } is constructed with merging two same copies of directed cycle graph with
indefinite path length . We denote W = {r-Cm : Pr } as path union of r same copies of directed cycle
graph.
3. Results
Theorem 3.1. The linear directed cycle Cm is directed cordial.
Proof: Let t1, t2, . . . tm be successive vertices of directed cycle Cm.
Consider a function 𝜌: V (W) → {0, 1} resulted as following
Case I For m ≡ 0, 1 (mod4)
0 ; 𝑝 = 0, 3 (mod 4)
𝜌(tp) = { 1 ; 𝑝 = 1, 2(mod 4)
Case II For m ≡ 2, 3 (mod4)
0 ; p = 0, 2 (mod4)
𝜌(tp) = { 1 ; p = 1, 3(mod4)
Table 1: Cordial pattern for vertices and edges
Conditions on m Vertices Pattern Edges Pattern
m≡ 0(mod4) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1)
m ≡ 1(mod4) 𝑣𝜌 (0)+1 =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1) +1
m ≡2(mod4) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1)
29
m ≡3(mod4) 𝑣𝜌 (0)+1 =𝑣𝜌 (1) 𝑒𝜌 (0)+1 =𝑒𝜌 (1)
The corresponding observed cases of m with labeling pattern of edges and vertices resulted in
above table. Therefore, the conditions |𝑒𝜌 (0) −𝑒𝜌 (1)| ≤ 1 and |𝑣𝜌 (0) −𝑣𝜌 (1)| ≤ 1 are preserved and
verified.
Example 1. The directed cordiality of C6 is elaborated as shown in Figure 1.
Figure 1: 𝐶6 is directed cordial
Theorem 3.2. The linear directed square cycle Cm2, m ≥ 6 is directed cordial.
Proof: Let t1, t2, . . . tm successive vertices of directed square cycle Cm.
Consider a function 𝜌 : V (W) → {0, 1} resulted as following
For m ≡ 0, 2, 4 (mod6)
1 ; p = 1, 3, 5(mod6)
𝜌(tp) = { 0 ; p = 0, 2, 4(mod6)
Table 2: Cordial pattern for vertices and edges
Conditions on m Vertices Pattern Edges Pattern
m≡ 0,2, 4(mod6) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1)
The corresponding observed cases of m with labeling pattern of edges and vertices resulted in
above table. Therefore, the conditions |𝑒𝜌 (0) −𝑒𝜌 (1)| ≤ 1 and |𝑣𝜌 (0) −𝑣𝜌 (1)| ≤ 1 are preserved and
verified.
Example 2. The directed cordiality C62 is elaborated as shown in Figure 2.
Figure 2: C62 is directed cordial
Theorem 3.3. The graph W = {2-Cm : Pr }is directed cordial.
30
Proof: Let us denote 1st and 2nd copies of m-pan as t1, t2, . . .tm and s1, s2 . . . sm respectively. Let the
vertices 1st, 2nd and r th of path Pr be represented by x1, x2, . . . , xr which carries condition as first
vertex x1 = t1 and r th vertex, xr = s1.
Consider a mapping : V (W) → {0, 1} as given below
Some cases observed
Case I For r ≡ 0(mod4), m ≡ 0, 1 (mod4)
0 ; p = 0, 3(mod4)
𝜌(tp) = {
1 ; p = 1, 2(mod4)
0 ; p = 1, 2(mod4)
𝜌(sp) = {
1 ; p = 0, 3 (mod4)
0 ; p = 0, 3(mod4)
𝜌(xp) = {
1 ; p = 1, 2(mod4)
Case II For r ≡ 1(mod4), m ≡ 0 (mod4)
0 ; p = 0, 3(mod4)
𝜌(tp) = 𝜌(sp) = {
1 ; 𝑝 = 1, 2(𝑚𝑜𝑑4)
0 ; p = 0, 3(mod4)
𝜌(xp) = {
1 ; 𝑝 = 1, 2(𝑚𝑜𝑑4)
Case III For r ≡ 2 (mod4), m ≡ 0, 1 (mod4)
0 ; p = 0, 3(mod4)
𝜌(tp) = {
1 ; p = 1, 2(mod4)
0 ; p = 1, 2(mod4)
𝜌(sp) = {
1 ; p = 0, 3 (mod4)
0 ; p = 0, 2(mod4)
𝜌(xp) = {
1 ; p = 1, 3(mod4)
Case IV For r ≡ 3 (mod4), m ≡ 0, 1 (mod4)
0 ; p = 0, 3(mod4)
𝜌(tp) = {
1 ; p = 1, 2(mod4)
0 ; p = 1, 2(mod4)
𝜌(sp) = {
1 ; p = 0, 3 (mod4)
0 ; p = 2, 3(mod4)
𝜌(xp) = {
1 ; p = 0, 1(mod4)
Case V For r ≡ 0(mod4), m ≡ 2, 3 (mod4)
0 ; p = 0, 2(mod4)
𝜌(tp) = {
1 ; p = 1, 3(mod4)
0 ; p = 1, 3(mod4)
𝜌(sp) = {
1 ; p = 0, 2 (mod4)
31
0 ; p = 0, 3(mod4)
𝜌(xp) = {
1 ; p = 1, 2(mod4)
Case VI For r ≡ 1(mod4), m ≡ 2 (mod4)
0 ; p = 0, 3(mod4)
𝜌(tp) = 𝜌(sp) = {
1 ; 𝑝 = 1, 2(𝑚𝑜𝑑4)
0 ; p = 0, 3(mod4)
𝜌(xp) = {
1 ; 𝑝 = 1, 2(𝑚𝑜𝑑4)
Case VII For r ≡ 2 (mod4), m ≡ 2, 3 (mod4)
0 ; p = 0, 2(mod4)
𝜌(tp) = {
1 ; p = 1, 3(mod4)
0 ; p = 1, 3(mod4)
𝜌(sp) = {
1 ; p = 0, 2 (mod4)
0 ; p = 0, 2(mod4)
𝜌(xp) = {
1 ; p = 1, 3(mod4)
Case VIII For r ≡ 3 (mod4), m ≡ 2, 3 (mod4)
0 ; p = 0, 2(mod4)
𝜌(tp) = {
1 ; p = 1, 3(mod4)
0 ; p = 1, 3(mod4)
𝜌(sp) = {
1 ; p = 0, 2 (mod4)
0 ; p = 2, 3(mod4)
𝜌(xp) = {
1 ; p = 0, 1(mod4)
Table 3: Cordial pattern for vertices and edges
Conditions on r, m Vertices Pattern Edges Pattern
r≡ 0(mod4), m ≡ 0,1,2,3(mod4) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1) +1
r ≡ 0(mod4), m ≡ 0,2(mod4) 𝑣𝜌 (0) =𝑣𝜌 (1) +1 𝑒𝜌 (0) =𝑒𝜌 (1)
r ≡0(mod4), m ≡ 0,1,2,3(mod4) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1) +1
r ≡0(mod4), m ≡ 0,1,2,3(mod4) 𝑣𝜌 (0) =𝑣𝜌 (1) +1 𝑒𝜌 (0) =𝑒𝜌 (1)
The corresponding observed cases of m with labeling pattern of edges and vertices resulted in
above table. Therefore, the conditions |𝑒𝜌 (0) −𝑒𝜌 (1)| ≤ 1 and |𝑣𝜌 (0) −𝑣𝜌 (1)| ≤ 1 are preserved and
verified.
Example 3 The graph W = {2-C5 : P6 } elaborated in Figure 3 is directed cordial.
32
Figure 3: The graph W = {2-C5 : P6 } is directed cordial
Theorem 3.4. The graph W = {r-Cm : Pr }is directed cordial.
Proof. Consider the vertices of 1st, 2nd , . . . , r th copy graph as tp1, tp2, . . . , tpr. Let x1, x2, . . . , xr be
the vertices of path Pr with condition x1 = tp1, x2 = tp2, . . . , xr = tpr.
Consider : V (W) → {0, 1} with following
We observe following cases
Case I: For r ≡ 0(mod4), m ≡ 0, 1 (mod4)
0 ; p = 1, 2(mod4)
𝜌(tpq) = {
1 ; 𝑝 = 0, 3(𝑚𝑜𝑑4)
0 ; p = 0, 3(mod4)
𝜌(tpq−2) = 𝜌(tpq−3) = {
1 ; p = 1, 2(mod4)
Case II: For r ≡ 1(mod4), m ≡ 0, 1 (mod4)
0 ; p = 0, 3(mod4)
𝜌(tpq) = {
1 ; 𝑝 = 1, 2(𝑚𝑜𝑑4)
0 ; p = 1, 2(mod4)
𝜌(tpq−1) = 𝜌(tpq−2) = {
1 ; p = 0, 3(mod4)
Case III: For r ≡ 2(mod4), m ≡ 0, 1 (mod4)
0 ; p = 0, 3(mod4)
𝜌(tpq) = {
1 ; 𝑝 = 1, 2(𝑚𝑜𝑑4)
0 ; p = 1, 2(mod4)
𝜌 (tpq−2) = 𝜌 (tpq−4) = {
1 ; 𝑝 = 0, 3(𝑚𝑜𝑑4)
Case IV: For r ≡ 3(mod4), m ≡ 0, 1 (mod4)
0 ; 𝑝 = 0, 3(𝑚𝑜𝑑4)
𝜌(tpq) = {
1 ; 𝑝 = 1, 2(𝑚𝑜𝑑4)
0 ; p = 1, 2(mod4)
𝜌(tpq−1) = 𝜌(tpq−4) = {
1 ; 𝑝 = 0, 3(𝑚𝑜𝑑4)
Case V: For r ≡ 0(mod4), m ≡ 2, 3 (mod4)
33
0 ; p = 1, 3(mod4)
𝜌(tpq) = {
1 ; 𝑝 = 0, 2(𝑚𝑜𝑑4)
0 ; p = 0, 2(mod4)
𝜌(tpq−2) = 𝜌(tpq−3) = {
1 ; p = 1, 3(mod4)
Case VI: For r ≡ 1(mod4), m ≡ 2, 3 (mod4)
0 ; p = 0, 2(mod4)
𝜌(tpq) = {
1 ; 𝑝 = 1, 3(𝑚𝑜𝑑4)
0 ; p = 1, 3(mod4)
𝜌(tpq−1) = 𝜌(tpq−2) = {
1 ; p = 0, 2(mod4)
Case VII: For r ≡ 2(mod4), m ≡ 2, 3 (mod4)
0 ; 𝑝 = 0, 2(𝑚𝑜𝑑4)
𝜌(tpq) = {
1 ; 𝑝 = 1, 3(𝑚𝑜𝑑4)
0 ; p = 1, 3(mod4)
𝜌(tpq−2) = 𝜌(tpq−4) = {
1 ; 𝑝 = 0, 2(𝑚𝑜𝑑4)
Case VIII: For r ≡ 3(mod4), m ≡ 2, 3 (mod4)
0 ; 𝑝 = 0, 2(𝑚𝑜𝑑4)
𝜌(tpq) = {
1 ; 𝑝 = 1, 3(𝑚𝑜𝑑4)
0 ; p = 1, 3(mod4)
𝜌(tpq−1) = 𝜌(tpq−4) = {
1 ; 𝑝 = 0, 2(𝑚𝑜𝑑4)
Table 4: Cordial pattern for vertices and edges
Conditions on r, m Vertices Pattern Edges Pattern
r ≡ 0(mod4),m ≡ 0, 2 (mod4) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1) +1
r ≡ 1(mod4), m ≡ 0, 2 (mod4) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1)
r ≡ 2(mod4), m ≡0, 2 (mod4) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1) +1
r ≡ 3(mod4), m ≡ 0, 2 (mod4) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1)
r ≡ 0(mod4),m ≡ 1, 3 (mod4) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1)+1
r ≡ 1(mod4),m ≡ 1, 3 (mod4) 𝑣𝜌 (0)+1 =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1) +1
r ≡ 2(mod4),m ≡ 1, 3 (mod4) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1) +1
r ≡ 3(mod4),m ≡ 1, 3 (mod4) 𝑣𝜌 (0) =𝑣𝜌 (1) +1 𝑒𝜌 (0)+1 =𝑒𝜌 (1)
The corresponding observed cases of m with labeling pattern of edges and vertices resulted in
above table. Therefore, the conditions |𝑒𝜌 (0) −𝑒𝜌 (1)| ≤ 1 and |𝑣𝜌 (0) −𝑣𝜌 (1)| ≤ 1 are preserved and
verified.
Example 4: The graph W = {5-Cm : P5 }is directed cordial as shown in Figure 4.
34
Figure 4: The graph W = {5-Cm : P5 }is directed cordial
Theorem 3.5. The graph W = {2-Cm2 : Pr } merged by two copies of directed cycle graph with path
of indefinite length is cordial.(For m ≥ 6)
Proof. Consider vertices of 1st and 2nd copies of Cm2 as t1, t2, . . .tm and s1, s2 . . . sm respectively.
Let vertices 1st, 2nd and r th of path Pr be represented by x1, x2, . . . , xr which carries condition as first
vertex x1 = t1 and r th vertex, xr = s1.
Consider : V (W) → {0, 1} as stated,
When m ≡ 0, 2, 4(mod6), we observe same cases given below
Case I For r ≡ 0 (mod4)
1 ; p = 1, 3, 5(mod6)
𝜌(tp) = { 0 ; p = 0, 2, 4(mod6)
1 ; p = 0, 2, 4(mod6)
𝜌(sp) = { 0 ; p = 1, 3, 5(mod6)
1 ; r = 1, 2(mod4)
𝜌(xr) ={
0 ; r = 0, 3(mod4)
Case II For r ≡ 1(mod4)
1 ; p = 1, 3, 5(mod6)
𝜌(tp) = 𝜌(sp) = { 0 ; p = 0, 2, 4(mod6)
1 ; r = 1, 2(mod4)
𝜌(xr) ={
0 ; r = 0, 3(mod4)
Case III For r ≡ 2, 3 (mod4)
1 ; p = 1, 3, 5(mod6)
𝜌(tp) = { 0 ; p = 0, 2, 4(mod6)
1 ; p = 0, 2, 4(mod6)
𝜌(sp) = { 0 ; p = 1, 3, 5(mod6)
1 ; r = 0, 1(mod4)
𝜌(xr) = {
0 ; r = 2, 3(mod4)
Table 5: Cordial pattern for vertices and edges
35
Conditions on r, m Vertices Pattern Edges Pattern
r ≡ 0(mod4),m ≡ 0, 2, 4 (mod6) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1) +1
r ≡ 1(mod4), m ≡ 0, 2, 4(mod6) 𝑣𝜌 (0) =𝑣𝜌 (1) +1 𝑒𝜌 (0) =𝑒𝜌 (1)
r ≡ 2(mod4), m ≡ 0, 2, 4(mod6) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1) +1
r ≡ 3(mod4), m ≡ 0, 2, 4(mod6) 𝑣𝜌 (0) =𝑣𝜌 (1) +1 𝑒𝜌 (0) =𝑒𝜌 (1)
The corresponding observed cases of r, m with labeling pattern of edges and vertices resulted in
above table. Therefore, the conditions |𝑒𝜌 (0) −𝑒𝜌 (1)| ≤ 1 and |𝑣𝜌 (0) −𝑣𝜌 (1)| ≤ 1 are preserved and
verified.
Example 5 The graph W = {2-C62 : P5 } is directed cordial as shown in Figure 5.
Figure 5: The graph W = {2-C62 : P5 } is directed cordial
Theorem 3.6. The graph W = {r-Cm2 : Pr }, m ≥ 6 preserves directed cordial labeling.
Proof. Let us represent the vertices of 1st, 2nd , . . . , r th copy as tp1, tp2, . . . , tpr. Let x1, x2, . . . , xr be
the vertices of path Pr with condition x1 = tp1, x2 = tp2, . . . , xr = tpr.
Consider 𝜌 : V (W) → {0, 1} When m ≡ 0, 2, 4(mod6).
We examine same cases.
Case I For r ≡ 0(mod4)
1 ; 𝑝 = 0, 2, 4(mod6)
𝜌(tpq) = { 0 ; 𝑝 = 1, 3, 5(mod6)
1 ; 𝑝 = 1, 3, 5(mod6)
𝜌(tpq−2) = 𝜌(tpq−3) = { 0 ; 𝑝 = 0, 2, 4(mod6)
Case II For r ≡ 1(mod4)
1 ; 𝑝 = 1, 3, 5(mod6)
𝜌(tpq) = { 0 ; ℎ = 0, 2, 4(mod6)
1 ; p = 0, 2, 4(mod6)
𝜌(tpq−1) = 𝜌(tpq−2) = { 0 ; p = 1, 3, 5(mod6)
Case III For r ≡ 2 (mod4)
36
1 ; p = 1, 3, 5(mod6)
𝜌(tpq) = { 0 ; p = 0, 2, 4(mod6)
1 ; p = 0, 2, 4(mod6)
𝜌(tpq−2) = 𝜌(tpq−4) = { 0 ; p = 1, 3, 5(mod6)
Case IV For r ≡ 3 (mod4)
1 ; p = 1, 3, 5(mod6)
𝜌(tpq) = { 0 ; p = 0, 2, 4(mod6)
1 ; p = 0, 2, 4(mod6)
𝜌(tpq−1) = 𝜌(tpq−4) = { 0 ; p = 1, 3, 5(mod6)
Table 6: Cordial pattern for vertices and edges
Conditions on r, m Vertex Pattern Edges Pattern
r ≡ 0(mod4),m≡ 0, 2, 4 (mod6) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1) +1
r ≡ 1(mod4), m≡ 0, 2, 4 (mod6) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1)
r≡ 2(mod4),m≡ 0, 2, 4 (mod6) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1) +1
r ≡ 3(mod4), m≡ 0, 2, 4 (mod6) 𝑣𝜌 (0) =𝑣𝜌 (1) 𝑒𝜌 (0) =𝑒𝜌 (1)
The corresponding observed cases of r, m with labeling pattern of edges and vertices resulted in
above table. Therefore, the conditions |𝑒𝜌 (0) −𝑒𝜌 (1)| ≤ 1 and |𝑣𝜌 (0) −𝑣𝜌 (1)| ≤ 1 are preserved and
verified.
Example 6 The graph W = {6-C62 : P6 } is directed cordial as shown in Figure 6.
Figure 6: The graph W = {6-C62 : P6 } is cordial
4. Conclusion
In this paper we investigated linear directed cycle and their square is directed cordial. The directed
path merged with two copies of directed cycle Cm and directed square cycle Cm2 is directed cordial.
We proved that directed path unions of r-copies of these graphs are directed cordial under certain
condition. To investigate and elaborate various families of graph which preserves same results is an
open problem for researchers. In the branch of graph theory, labeling is widely applicable in coding,
circuit designing, communication networking and data base management.
37
5. Acknowledgements
Authors are highly grateful to anonymous referee for their valuable inputs and comments.
6. References
[1] Cahit, Cordial graphs: A Weaker version of graceful and harmonious graphs, Ars combinatorica,
23 (1987), 201–207.
[2] I.Cahit, On cordial and 3-equitable labelings of graphs, Utilitas Math, 370 (1990), 189– 198.
[3] J.A. Gallian 2016. A Dynamic survey of graph labeling, The Electronics Journal of
Combinatorics, (2016), DS6.
[4] F.Harary, Graph Theory, Addison-Wesley, Reading, Massachusetts,(1972).
[5] Al-Shamiri, M.M.A., Nada, S.I., Elrokh, A.I. and Elmshtaye, Y. Some Results on Cordial
Digraphs. Open Journal of Discrete Mathematics, (2020), 10, 4-12.
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