Let r be any natural number. An injective function , where  is the Gaussian Tribonacci number in the Gaussian Tribonacci sequence is said to be Gaussian Tribonacci r-graceful labeling if the induced edge labeling such that  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 , 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

Sheriba, M

Sunitha, K

English

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Ratio Mathematica                                                                            Volume 44, 2022   Gaussian Tribonacci R-Graceful Labeling of Some Tree Related Graphs  Dr. Sunitha. K1  Sheriba. M2  Abstract Let r be any natural number. An injective function kallforGTkikikiGV rq }...,,2,2,1,1,,0{)(: 1 , where 1rqGT  is the thrq )1(  Gaussian Tribonacci number in the Gaussian Tribonacci sequence is said to be Gaussian Tribonacci r-graceful labeling if the induced edge labeling },...,,{)(: 121* rqGTGTGTGE 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 ][ ,nmBS  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:05C783                                                               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.  188Dr. 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rqGT is thethrq )1(   Gaussian Tribonacci number in the Gaussian Tribonacci sequence is said to be Gaussian Tribonacci graceful labeling if the induced edge labeling}...,,,{)(: 121* rqGTGTGTGE 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 1KPn is obtained by joining a single pendent edge to each vertex of the path nP.  Definition 1.3 [3] A regular caterpillar graph 1nKPm  is obtained from the path mP by joining 1nK vertices to each vertices of the path mP.  189Gaussian Tribonacci R-Graceful Labeling of Some Tree Related Graphs  Definition 1.4 [8] The Bistar graph nmB , is obtained from 2K by attaching m pendent edges to one end of 2K  and n pendent edges to the other end of 2K .  Definition 1.5 [10] The Subdivision of Bistar graph )( ,nmBS is obtained by subdividing each edge of a Bistar graph nmB , .  Definition 1.6[9] A Coconut Tree graph CT(m,n) is obtained from the path nP by appending n new pendent edges at an end vertex of nP.  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 irqiii   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+4i0 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 190Dr. 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 rirqiii   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+2i7+4i 2+i7+4i 2+i0 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+13i1+i 2+i 4+2i 7+4i 13+7i1+i 147+80i 64+35i 105+57ii 75+41i149+81i 68+37i 112+61i088+48i44+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 thm  vertex of the path mP . The resultant graph is CT (m, n) whose vertex set is }}1/{}1/{{)],([ nivmiunmCTV ii    and edge set 191Gaussian 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 irqiii   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+7i7+4i4+2i2+i1+i10 81+44i 37+20i 61+33i 48+26i41+22i44+24i46+25i47+26i41+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 irqiii   1,1,)( )2(11   rnjGTv mjrqj  1,1,2,)()( )2()1(1   rnjmiGTvv mjnirqiij   Thus  admits Gaussian Tribonacci r-graceful labeling for all r . 192Dr. 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  0504+274i 274+149i504+274i 230+125i149+81i 379+206i2+i1+i377+205i 378+205i4+2i7+4i13+7i24+13i44+24i81+44i81+44i 44+24i 223+121i 226+123i480+261i 491+267i Figure 2.5  Theorem 2.5 The Bistar graph nmB ,  is Gaussian Tribonacci arbitrarily graceful for all .2, nm  Proof. Let vu, be the vertices of 2K and let miui 1, be the m vertices attached to one end of 2K and njv j 1, be the n vertices attached to the other end of 2K .The resultant graph isnmB ,  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. HencenmB ,  is Gaussian Tribonacci arbitrarily graceful for all .2, nm    Example 2.5 The Gaussian Tribonacci 1-graceful labeling of Bistar graph4,2B  is given in Figure 2.6 193Gaussian Tribonacci R-Graceful Labeling of Some Tree Related Graphs  13+7i7+4i4+2i2+i1+i1024+13i24+13i11+6i17+9i20+11i22+12i1+i1 Figure 2.6  Theorem 2.6 The subdivision of the bistar graph )( ,nmBS is Gaussian Tribonacci arbitrarily graceful for all 1, nm  Proof. Let vu, be the central vertices of the Bistar graph nmB , 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 )( ,nmBS  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)( 1121   rmiGTuuGTurGTvs irirqrq   1,1,)()(,1,1,)()( 2121   rniGTvvrmiGTuu irqiirmii   1,1,)()( 121   rniGTvv rimii   Thus   admits Tribonacci r-graceful labeling for all r . Hence the subdivision of the bistar graph )( ,nmBS is Gaussian Tribonacci arbitrarily graceful for all 1, nm .  Example 2.6 The Gaussian Tribonacci 1-graceful labeling of Subdivision of Bistar  graph S(2,2B ) is given in Figure 2.7 194Dr. Sunitha. K and sheriba. M  80+44i4+2i76+42i2+1i78+42i 80+43i1+1i1 81+44i81+44i0149+81i149+81i44+24i 105+57i7+4i112+61i24+13i125+68i13+7i138+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 13th 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). [7] Steven K lee, HunterLehmann, Andrew Park, Prime labeling of families of trees with Gaussian integers, AKCE International Journal of Graphs and Combinatorics 13 (2016), 165-176 [8] Thirugnanasambandan, K and Chitra G.,” Fibonacci Mean Anti-Magic Labeling of graphs”, International Journal of Computer Applications (0975-8887), Vol.134, No,15, January 2016.  [9] Uma. R and Amuthavalli. D, “Fibonacci graceful labeling of some star related graphs”, International Journal of Computer Applications (0975-8887) Vol.134, No.15, January 2016. 195Gaussian 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).  [11] Yuksel Soykan, Ekan Tasdemir, Inci Okumus, Melih Gocen,” Gaussian Generalized  Tribonacci Numbers”, Journal of Progressive Research in Mathematics (JPRM), Vol.14, Issue:2. 196

Gaussian Tribonacci R-Graceful Labeling of Some Tree Related Graphs

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Gaussian Tribonacci R-Graceful Labeling of Some Tree Related Graphs

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