Ratio Mathematica                                                                            Volume 44, 2022 

 
 

Gaussian Tribonacci R-Graceful Labeling of 

Some Tree Related Graphs 

 
Dr. Sunitha. K

1
 

 Sheriba. M
2 

 
Abstract 

Let r be any natural number. An injective function 

kallforGTkikikiGV
rq

}...,,2,2,1,1,,0{)(:
1

 , where 
1rq

GT  is the 

th
rq )1(  Gaussian Tribonacci number in the Gaussian Tribonacci sequence is said to 

be Gaussian Tribonacci r-graceful labeling if the induced edge labeling 

},...,,{)(:
121

*




rq
GTGTGTGE such that |)()(|)(

*
vuuv    is bijective. If a graph 

G admits Gaussian Tribonacci r-graceful labeling, then G is called a Gaussian 

Tribonacci r-graceful graph. A graph G is said to be Gaussian Tribonacci arbitrarily 

graceful if it is Gaussian Tribonacci r-graceful for all r. In this paper we investigate the 

Path graph nP , the Comb graph 1KPm , the Coconut tree graph ,),( nmCT the regular 

caterpillar graph 1nKPm , the Bistar graph nmB ,  and the Subdivision of Bistar 

graph ][
,nm

BS
 
are Gaussian Tribonacci arbitrarily graceful. 

 

Keywords:  Gaussian Tribonacci sequence, Gaussian Tribonacci graceful labeling, 

Path graph, Comb graph, Coconut Tree graph, Regular caterpillar graph, Bistar graph 

and Subdivision of Bistar graph. 

 

Subject Classification:05C78
3
 

 

 

 

 

                                                           
1
 Assistant Professor, Department of Mathematics, Scott Christian College (Autonomous), Nagercoil-

629003 

Email: ksunithasam@gmail.com  
2
 Part Time Research Scholar, Department of Mathematics, Scott Christian College (Autonomous), 

Nagercoil-629003.Affiliated to Manonmaniam Sundaranar University, Tirunelveli-627012 

Email: sheribajerin@gmail.com    
3
 Received on June 19th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 

10.23755/rm.v44i0.906. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published 

under the CC-BY licence agreement. 

 

188

mailto:ksunithasam@gmail.com
mailto:sheribajerin@gmail.com


Dr. Sunitha. K
 
and sheriba. M

 

 

 

1. Motivation and Main Results 

 Graphs considered throughout this paper are finite, simple, undirected and nontrivial. 

Labeling of graph is an assignment of values to vertices and edges or both subject to 

certain conditions. In 1967, Rosa [6] introduced the concept of graceful labeling. In 

1982, Slater [4] introduced the concept of k-graceful labeling of graphs and is defined 

as follows: Let G be a simple graph with p vertices and q edges. Let k be any natural 

number. Define an injective mapping }1...,,2,1,0{)(:  kqGV  that induces bijective 

mapping }1,,....1,{)(:
*

 kqkkGE
 

where |)()(|)(
*

vuuv   for all 

)(GEuv  and ),(, GVvu   then   is called k-graceful labeling while 
*

  is called an 

induced edge k-graceful labeling and the graph G is called k-graceful graph. Graphs that 

are k-graceful for all k are sometimes called arbitrarily graceful. Yuksel Soykan e tal. 

[11] introduced the concept of Gaussian Generalised Tribonacci Numbers. In this 

sequel, we introduce a new concept Gaussian Tribonacci r-graceful labeling of graphs. 

We follow D. B. West [10] and J. A. Gallian [2], for standard terminology and 

notations. 

 

Definition 1.1 Let G be a graph with p vertices and q edges. Let r be any natural 

number. An injective function 

kallforGTkikikiGV
rq

}...,,,2,1,1,,0{)(:
1


,where 1rq

GT
 is the

th
rq )1( 

 

Gaussian Tribonacci number in the Gaussian Tribonacci sequence is said to be Gaussian 

Tribonacci graceful labeling if the induced edge 

labeling
}...,,,{)(:

121

*




rq
GTGTGTGE

such that 
|)()(|)(

*
vuuv  

 is bijective. 

If a graph G admits Gaussian Tribonacci graceful labeling, then G is called a Gaussian 

Tribonacci graceful graph. A graph G is said to be Gaussian Tribonacci arbitrarily 

graceful if it is Gaussian Tribonacci r-graceful for all r. 

 

Remark 1.1 [11] The Gaussian Tribonacci sequence is obtained as follows: 

31,1,0
321210




nGTGTGTGTandiGTGTGT
nnnn  

...},1324,713,47,24,2,1,1,0{, iiiiiiie  is the Gaussian Tribonacci 

sequence. 

 

Definition 1.2 [9] The Comb graph 1
KP

n


is obtained by joining a single pendent edge 

to each vertex of the path n
P

. 

 

Definition 1.3 [3] A regular caterpillar graph 1
nKP

m


 is obtained from the path m
P

 by 

joining 1
nK

vertices to each vertices of the path m
P

. 

 

189



Gaussian Tribonacci R-Graceful Labeling of Some Tree Related Graphs 
 

Definition 1.4 [8] The Bistar graph nm
B

, is obtained from 2
K

by attaching m pendent 

edges to one end of 2
K

 and n pendent edges to the other end of 2
K

. 

 

Definition 1.5 [10] The Subdivision of Bistar graph )(
,nm

BS is obtained by subdividing 

each edge of a Bistar graph 
nm

B
,

. 

 

Definition 1.6[9] A Coconut Tree graph CT(m,n) is obtained from the path n
P

 by 

appending n new pendent edges at an end vertex of n
P

. 

 

2. Main Results 
Theorem 2.1 The path graph nP  is Gaussian Tribonacci arbitrarily graceful for 

all .2n  
Proof. 

Let nP  be a path graph of length 1n  with vertex set }1/{)( niuPV in   and edge 

set }11/{)( 1   niuuPE iin  such that 1|)(||)(|  nqPEandnpPV nn  
Define kallforGTkikikiPV

rqn
}...,,2,2,1,1,,0{)(:

1
  by 

01
)( Tu 

 
1,2,)1()()(

11



rniGTuu

irq

i

ii
  

Thus  admits Gaussian Tribonacci graceful labeling for all r . 

Hence the path graph nP  is Gaussian Tribonacci arbitrarily graceful for all 2n . 

 

Example 2.1 

The Gaussian Tribonacci 2-graceful labeling of path graph 5P  is given in Figure 2.1 
1+i2+i4+2i7+4i

0 7+4i 3+2i 5+3i 4+2i
 

Figure 2.1 

 

Theorem 2.2 The Comb graph 1KPn  is Gaussian Tribonacci arbitrarily graceful for all 

2n . 

Proof. 

Let niui 1, be the vertices of the path nP  and let nivi 1, be the vertices which 

are joined to each vertices of the path nP .The resultant graph 1KPn whose vertex set 

is }1/,{)( 1 nivuKPV iin  and edge set 

is }}11/{}1/{{)( 11   niuunivuKPE iiiin  
such that npKPV n 2|)(| 1   

and 12|)(| 1  nqKPE n  

Case 1 

190



Dr. Sunitha. K
 
and sheriba. M

 

 

For 2n  
Define kallforGTkikikiPV

rqn
}...,,2,2,1,1,,0{)(:

1
  by 

1,)(,)(
1201




rGTuGTu
r

 , 

1,)()(,1,)(
2211




rGTuvrGTv
rrq

  

Case 2 

For 3n  
Define kallforGTkikikiPV

rqn
}...,,2,2,1,1,,0{)(:

1
  by  

01
)( GTu  , 1,)(,1,2,)1()()( 111   rGTvrniGTuu rirq

i

ii
  

1,2,)()(
1




rniGTuv
irii

  

Thus
 
admits Gaussian Tribonacci graceful labeling for all r. 

Hence the comb graph 1KPn is Gaussian Tribonacci arbitrarily graceful for all 2n . 

 

Example2.2 

The Gaussian Tribonacci 3-graceful labeling of Comb graph 
12

KP   is given in Figure 

2.2 

4+2i

7+4i 2+i

7+4i 2+i

0 4+2i

 
Figure 2.2 

The Gaussian Tribonacci 2-graceful labeling of Comb graph 15 KP   is given in Figure 

2.3 

 

149+81i 81+44i 24+13i

1+i 2+i 4+2i 7+4i 13+7i

1+i 147+80i 64+35i 105+57ii 75+41i

149+81i 68+37i 112+61i0
88+48i

44+24i

 
Figure 2.3 

 

Theorem 2.3 The Coconut Tree graph CT (m, n) is Gaussian Tribonacci arbitrarily 

graceful for all .2, nm  

Proof. Let miui 1,  be the vertices of the path mP  and let nivi 1,  be the new 

vertices which are attached to the 
th

m  vertex of the path mP . The resultant graph is CT 

(m, n) whose vertex set is }}1/{}1/{{)],([ nivmiunmCTV ii    and edge set 

191



Gaussian Tribonacci R-Graceful Labeling of Some Tree Related Graphs 
 

is }}1/{}11/{{)],([ 1 nivumiuunmCTE imii    such that 

nmpnmCTV |)],([|  and 1|)],([|  nmqnmCTE  

Define kallforGTkikikiPV
rqn

}...,,2,2,1,1,,0{)(:
1

  by  

,)(
01

GTu   

1,2,)1()()(
11




rmiGTuu
irq

i

ii
  

1,1,)()(
1




rniGTuv
irmi

  

Thus   admits Gaussian Tribonacci graceful labeling for all r. 

Hence the Coconut Tree graph CT(m,n) is Gaussian Tribonacci arbitrarily graceful for 

all 2, nm . 

 

Example 2.3 

The Gaussian Tribonacci1-graceful labeling of Coconut Tree graph )5,5(CT  is given in 

Figure 2.4 

81+44i 44+24i 24+13i 13+7i

7+4i

4+2i

2+i

1+i
1

0 81+44i 37+20i 61+33i 48+26i

41+22i

44+24i

46+25i

47+26i

41+25i

`

 
Figure 2.4 

 

Theorem 2.4 The regular caterpillar graph 1nKPm  is Gaussian Tribonacci arbitrarily 

graceful for all 2, nm  

Proof.  

Let mivi 1, be the vertices of the path mP and let njmivij  1,1, be the 

vertices attached to each vertices of the path .mP The resultant graph is 1nKPm  
whose 

vertex set is }}1,1/{}1/{{][
1

njmivmivnKPV
ijim

   and edge set 

is }}1,1/{}11/{{][
11

njmivvmivvnKPE
ijiiim




 such 

that 1|][||][| 11  mnmqnKPEandmnmpnKPV mm  

Define a function kallforGTkikikiPV
rqn

}...,,2,2,1,1,,0{)(:
1

  by 

1,2,)1()()(,)(
1101




rmiGTvvGTv
irq

i

ii
  

1,1,)(
)2(11




rnjGTv
mjrqj

  

1,1,2,)()(
)2()1(1




rnjmiGTvv
mjnirqiij

  

Thus  admits Gaussian Tribonacci r-graceful labeling for all r . 

192



Dr. Sunitha. K
 
and sheriba. M

 

 

Hence the regular caterpillar graph 1nKPm  are Gaussian Tribonacci arbitrarily graceful 

for all .2, nm
 

 

Example 2.4 The Gaussian Tribonacci 2-graceful labeling of Regular caterpillar 

graph
14

2KP   is given in Figure 2.5 

 

0

504+274i 274+149i
504+274i 230+125i

149+81i 379+206i

2+i
1+i

377+205i 378+205i

4+2i7+4i

13+7i24+13i

44+24i81+44i

81+44i 44+24i 223+121i 226+123i

480+261i
491+267i

 
Figure 2.5 

 

Theorem 2.5 The Bistar graph 
nm

B
,

 is Gaussian Tribonacci arbitrarily graceful for all 

.2, nm
 

Proof. 

Let vu, be the vertices of 
2

K and let miui 1, be the m vertices attached to one end of 

2
K and njv

j
1, be the n vertices attached to the other end of 2K .The resultant 

graph is
nm

B
,  

whose vertex set is }1,1/,,,{)(
,

njmivuvuBV
jinm

 and edge set 

is }}{}1/{}1/{{)(
,

uvnjvvmiuuBE
jinm

   such that 

2|)(|
,

 nmpBV
nm

 and 1|)(|
,

 nmqBE
nm

 

Define a function kallforGTkikikiPV
rqn

}...,,2,2,1,1,,0{)(:
1

  by
 

1,1,)()(,)(,)(
110




rniGTvvGTvGTu
irqirq

  

1,1,)(
1




rmiGTu
inrqi

  

Thus   admits Gaussian Tribonacci r-graceful labeling for all r. 

Hence
nm

B
,

 is Gaussian Tribonacci arbitrarily graceful for all .2, nm
 

 

 

Example 2.5 

The Gaussian Tribonacci 1-graceful labeling of Bistar graph
4,2

B  is given in Figure 2.6 

193



Gaussian Tribonacci R-Graceful Labeling of Some Tree Related Graphs 
 

13+7i

7+4i

4+2i

2+i

1+i

1

0

24+13i

24+13i

11+6i

17+9i

20+11i

22+12i

1+i

1

 
Figure 2.6 

 

Theorem 2.6 The subdivision of the bistar graph )(
,nm

BS is Gaussian Tribonacci 

arbitrarily graceful for all 1, nm  

Proof. 

Let vu, be the central vertices of the Bistar graph 
nm

B
,

and let 

nivandmiu
ii

 1,1,
 

be the vertices joined with u and v respectively. 

Let nivandmius
ii

 1,1,,
11

 
be the new vertices obtained by subdividing the edges 

nivvandmiuuuv
ii

 1,1,,
 
respectively. The resulting graph is )(

,nm
BS  whose 

vertex set is 

}}{},{}1/,{}1/,{{)]([
11

,
svunivvmiuuBSV

iiiinm
 

 
and 

edge set 

is }}1/,{}{}{}1/,{{)]([
1111

,
nivvvvsvusmiuuuuBSE

iiiiiinm
 

 
such that 3)(2|)]([|

,
 nmpBSV

nm  
and 2)(2|)]([|

,
 nmqBSE

nm  
Define a function kallforGTkikikiPV

rqn
}...,,2,2,1,1,,0{)(:

1
 by 

1,1,)()()(,1,)(,0)(
1

1

21



rmiGTuuGTurGTvs

irirqrq


 
1,1,)()(,1,1,)()(

2

1

2

1



rniGTvvrmiGTuu

irqiirmii
  

1,1,)()(
12

1



rniGTvv

rimii
  

Thus   admits Tribonacci r-graceful labeling for all r . 

Hence the subdivision of the bistar graph )(
,nm

BS is Gaussian Tribonacci arbitrarily 

graceful for all 1, nm . 

 

Example 2.6 

The Gaussian Tribonacci 1-graceful labeling of Subdivision of Bistar  graph S(
2,2

B ) is 

given in Figure 2.7 

194



Dr. Sunitha. K
 
and sheriba. M

 

 
80+44i

4+2i

76+42i

2+1i

78+42i 80+43i

1+1i

1 81+44i

81+44i

0

149+81i
149+81i

44+24i 105+57i

7+4i

112+61i

24+13i

125+68i

13+7i

138+85i 
Figure 2.7 

 

 

3. Conclusion 
 

 In this paper, we investigate the path graph, the Comb graph, the Coconut Tree 

graph, the Regular caterpillar graph, the Bistar graph and the Subdivision of Bistar 

graph are Gaussian Tribonacci arbitrarily graceful. In future, we investigate Gaussian 

Tribonacci arbitrarily graceful labeling of cycle related graphs. 

 

References 
 

[1] David.W.and Anthoney. E. Barakaukas, “Fibonacci Graceful Graphs”. 

[2] J. A. Gallian, A Dynamic Survey of Graph Labeling, the Electronic Journal of 

Combinatorics, (2013). 

[3]  Murugesan. N and Uma. R, “Super vertex Gracefulness of Some Special Graphs”,  

IQSR Journal of Mathematics, Vol.:11, Issue 3 ver (may - june 2015), pp. 07-15 

[4] P. J. Slater, On k-graceful graphs, In: Proc. of the 13
th

 South Eastern Conference on 

 Combinatorics, Graph Theory and Computing (1982),53-57. 

[5] P. Prathan and Kamesh Kumar, “On k-Graceful Labeling of some graphs”, Journal 

of Applied Mathematics, Vol.: 34 (2016), No.1 – 2, pp. 09-17 

[6] Rosa. A “On Certain Valuation of vertices of graph”, (1967). 

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[9] Uma. R and Amuthavalli. D, “Fibonacci graceful labeling of some star related 

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195



Gaussian Tribonacci R-Graceful Labeling of Some Tree Related Graphs 
 

 [10]  West.D.B. B, Introduction to Graph Theory, Prentice-Hall of India, New Delhi, 

(2001). 

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196