Sciresol Sciresol https://indjst.org/author-guidelines Indian Journal of Science and Technology 0974-5645 10.17485/IJST/v15i14.254 research article Some New Classes of (k, d) Graceful 3 Distance Trees and 3 Distance Unicyclic Graphs Mohanty Gobind 1 Mishra Debdas 2 Sarangi Pravat 3 Bhattacharjee Subarna subarna.bhatt@gmail.com 1 Department of Mathematics, Ravenshaw University Cuttack, Odisha India Former Professor, C.V. Raman College of Engineering Bhubaneswar, Odisha India Department of Statistics, Ravenshaw University Cuttack, Odisha India 15 14 630 30 1 2022 4 3 2022 2022 Abstract

Objectives: To identify a new family of (k, d) graceful graphs. Methods: The methodology involves mathematical formulation for labeling of the vertices of a given graph and subsequently establishing that these formulations give rise to (k,d) graceful labeling. Findings: Here we define a three-distance tree as the tree possessing a path such that each vertex of the tree is at most at a distance three from that path. In this paper we identify two families of three distance trees that possess (k, d) graceful labeling. Furthermore, we show that the three distance unicyclic graphs obtained from these three distance trees by joining two end vertices of their central paths are also (k, d) graceful. Novelty: Here, we give (k, d) graceful labeling to two new families of graphs, namely some classes of three distance trees and three distance unicyclic graphs. This effort is the first of its kind which involves exploration of 3-distance (k, d) graceful graphs.

Keywords (k D) graceful labelling Hairy cycle Firecracker Three distance tree Three distance unicyclic graphs None
Introduction

Acharya and Hegde1 defined (k, d) − graceful labeling of a graph G with q edges as a surjective mapping of the vertex set of G into the set {0, 1, 2, ..., k + (q - 1)d} for some positive integers k and d. A (1, 1)-graceful labeling is called a graceful labeling and a (k, 1)-graceful labeling is called a k-graceful labeling. Bu and Zhang2 established that Km,n is (k, d)- graceful for all k and d and Kn is (k, d)-graceful if and only if k = d. Hegde and Shetty 3 showed that a tree T which can be transformed into a path by carrying out successive elementary transformations and the tree formed from T by subdividing each edge of T is (k, d)-graceful for all k and d. Some more results on graph labeling problems are found in some recent papers 4, 5, 6, 7, 8, 9, 10. For details of the literature involving (k, d) graceful graphs one may refer to the latest dynamic survey on graph labeling problems by Gallian 2.

From the literature survey it is found that there exist only some specific classes of graphs, namely  Km,n, Kn, and transformed trees which admit (k, d) graceful labeling. So, there is huge scope to explore in this area. In this paper we give (k, d)- graceful labeling to some new classes of three distance trees and three distance unicyclic graphs. Before deriving our results, we would like to have a recap of some of the existing graph theoretic terminologies and some new terminologies required for proving our results.

Definition 1.12 By a firecracker we mean a tree possessing a path known as the central path such that each vertex of the path is attached to the center of some star. Here we denote a firecracker by PnKmi,1,1,. mi 0,  i  =  1, 2, . . . , n,wheretheivertexofthepath Pnis attached to the center of the star Kmi,1.

Definition 1.2 A three distance tree T is a tree which contains a path H such that each vertex of T \ H is at a distance at most three from H.  We call the path H as the central path of T. Figure 1 represents a three-distance tree. A three distance unicyclic graph is a graph consisting of one cycle Cn such that each vertex of the graph is at distance at most three from Cn. Figure 2 represents a three distant unicyclic graph.

The three distance (k, d)-graceful trees in this paper are obtained by attaching leaves to the leaves and central path of firecrackers. The three distance unicyclic (k, d)- graphs in this paper are obtained by joining the end vertices of the central paths of three distance trees.

Here we use the method involving mathematical formulation for obtaining labeling of the vertices of a graph and then show that such a labeling is a (k ,d) - graceful labeling of that graph.

Results and Discussions

Constustion i 3.1 consider the fire cracker T  = PnKmi,1,1,.,  i  =  1, 2, . . . , n, whose vertices on the central Pnare c1, c2, c3, , cn. The vertices T \ Pnadjacent to ci are ci,i,  i  =   1, 2,  . . . ,  n T ci,i,are ci,i, ji,  ji   =   1, 2,  . . . ,  mi,  i   = 1, 2, . . . , n.Constructa 3-distance tree Tby attaching leaves to the vertices ci,i, jiand denote themby ci,i, ji, ti, ji,  ti, ji =  1, 2, . . . , si, ji,where si, jiis the number of leaves adjacent to ci,i, ji. All the vertices ci,i, jineed not be attached to leaves. Say, out of mivertices attached to ci,i,, riof the mattached to leaves. Let r  = max i=1nri.Assume that mi  rfor each i. Let ET= q. Obviously, q=2n-1+i=1nmi+jirisi, ji.

Theorem 3.1 The three distant trees in Construction 3.1 admit (k, d) graceful labeling with d  k.

Proof: Consider the three distant trees T in Construction 3.1. Define the mapping f:V (T ) - {0, 1, 2, 3, 4, , k + (q - 1)d} as follows.

For i = 1,2, ..., n, fci=i-1r+22d                    if i is oddk+q-i2r+2d    if i is even

and fci,i=k+q-i-1r+22-1d   if i is oddi2r+2-1d                      if i is even

For i = 1,2, ..., n,ji= 1,2, ..., mi,

fci, i, ji=jidifjirq+r-jidifji>rifi=1i-1r+22+jidifjirq-i-1r+22-p=1i-1mp-r+jp=1rsp, jp+r-jidifji>rifi is odd and i>1k+q-i-2r+22-1-jidifjirk+ir+22+p=1i-1mp-r+jp=1rsp, jp+ji-2difji>rifi is even

For i = 1,2, ..., n,ji= 1,2, ..., mi, ti, ji =  1, 2, . . . , si, ji, fci,i, ji, ti, ji=

k+m1-r+t1,1difji=1k+m1-r+z=1j1-1s1,z+j1-1+t1,j1difji>1ifi=1k+i-1r+22+p=1i-1mp-r+z=1jpsp,z+mi-r+ti,1difji=1k+i-1r+22+ji+p=1i-1mp-r+z=1jpsp,z+mi-r+z=1ji-1si,z+ti,ji-1difji>1ifi>1 is oddq-i-2r+22-p=1i-1mp-r+z=1jpsp,z-mi+r-ti,1-1difji=1q-i-2r+22-ji-p=1i-1mp-r+z=1jpsp,z-mi+r-z=1ji-1si,z-ti,jidifji>1ifi is even

Defining the labeling g on E(T ) by g(u, v)  =  |f (u) - f (v)|, we have

For i  =  1, 2, . . . , n-1,  gci, ci+1=  k + q - ir + 2d;

for i  =  1, 2, . . . , n, ji  =  1, 2, . . . , mi, ti, i, ji=  1, 2, . . . , si, i, ji,

gci,ci,i=k+q-i-1r+2-1d   if i is oddk+q-ir+2-1d                  if i is even; gci,i,ji, ci,i,ji,ti, i, ji              =k+m1-r+t1,1-1difji=1k+m1-r+z=1j1-1s1,z+t1,j1-1difji>1ifi=1k+p=1i-1mp-r+z=1jpsp, z+mi-r+ti,1-1difji=1k+p=1i-1mp-r+z=1jpsp, z+mi-r+z=1ji-1s1, z+ti, ji-1difji>1ifi>1  is odd

We find that g(u, v) assumes values {k, k + d, k + 2d, . . . , k + (q - 1)d}. Therefore, the labels of the edges of T constitute the set {k, k + d, k + 2d, . . . , k + (q - 1)d} and hence the mapping f is a (k, d) graceful labeling of T, i.e. T is a (k, d) graceful tree with d  k.

Theorem 3.2 The three distant unicyclic graphs obtained from the three distant trees in Construction 3.1 by joining the vertices c1 and cn admit (k, d) graceful labeling with d  k if n  (0 mod 4).

Proof: Consider the three distant unicyclic graph G in Theorem 3.2. Define the mapping

f : V (G) - {0, 1, 2, 3, 4, , k + (q - 1)d} as follows.

For i = 1,2, , n, pi=1, 2, , li,  ji=1, 2, , mi, ti,ji=1, 2, , si,ji,

fci= i-1r+22difi is odd and i<n2i-1r+22+1difi is odd and i>n2k+q-i2r+2difi is even;

fci,i= k+q-12i-1r+2-1ifi is odd ir+22-1difi is even and in2i2r+2difi is even and i>n2;

fci,i, ji= j1difjirq+r-j1difji>rifi=1i-1r+22+jidifjirq-i-1r+22-p=1i-1mp-r+jp=1rsp, jp+r-jiifji>rifi is odd and 1 <i <n2i-1r+22+ji+1difjirq-i-1r+22-p=1i-1mp-r+jp=1rsp, jp+r-jiifji>rifi is odd and i>n2k+q-i-2r+22-ji-1ifjirk+ir+22-p=1i-1mp-r+jp=1rsp, jp+ji-2ifji>rifi is even and 1 <i n2k+q-i-2r+22-ji-1ifjirk+ir+22-p=1i-1mp-r+jp=1rsp, jp+ji-1ifji>rifi is even and i>n2; fci,i, ji,ti,ji=  k+m1-r+t1,1difji=1k+m1-r+z=1j1-1s1,z+j1-1+t1,j1difji>1ifi=1k+i-1r+22+p=1i-1mp-r+z=1jpsp,z+mi-r+ti,1difji=1k+i-1r+22+ji+p=1i-1mp-r+z=1jpsp,z+mi-r+z=1ji-1si,z+ti,ji-1difji>1if1 <i <n2i is odd k+i-1r+22+p=1i-1mp-r+z=1jpsp,z+mi-r+ti,1+1difji=1k+i-1r+22+ji+p=1i-1mp-r+z=1jpsp,z+mi-r+z=1ji-1si,z+ti,jidifji>1ifi is odd and i>n2q-i-2r+22-p=1i-1mp-r+z=1jpsp,z-mi+r-ti,1difji=1q-i-2r+22-ji-p=1i-1mp-r+z=1jpsp,z-mi+r-z=1ji-1si,z-ti,jidifji>1ifi is even

Defining the labeling g on E(G) by g(u, v) = |f (u) − f (v)|, we have

for i=1,2,,n, gci, ci+1=k+q-ir+2difi<n2k+q-ir+2-1difin2;

for i=1,2,,n,ji=1,2,,mi,ti,i, ji=1,2,, si,i, ji

gci,ci,i=k+q-i-1r+2-1difi is odd and i<n2k+q-i-1r+2-2difi is odd and i>n2k+q-ir+2-1difi is even and in2k+q-ir+2difi is even and i>n2;

k+q-j1-1difjirk+j1-r-1difji>rifi=1k+q-i-1r+2-j1-1difjirk+p=1i-1mp-r+jp=1rsp,jp-r+ji-1difji>rifi is odd and i<n2k+q-i-1r+2-j1-2difjirk+p=1i-1mp-r+jp=1rsp,jp-r+ji-1difji>rifi is odd and i>n2k+q-i-1r+2-j1difjirk+p=1i-1mp-r+jp=1rsp,jp-r+ji-1difji>rifi is even and in2k+q-i-1r+2-j1-1difjirk+p=1i-1mp-r+jp=1rsp,jp-r+ji-1difji>rifi is even and i>n2

gci,i,ji,ci,i,ji,ti,ji= k+m1-r+t1,1 -1difjirk+m1-r+z=1j1-1s1,z+t1,j1-1difji>rifi=1k+p=1i-1mp-r+z=1jpsp,z+mi-r+ti,1-1difjirk+p=1i-1mp-r+z=1jpsp,z+mi-r+z=1ji-1s1,z+ti,ji-1difji>rifi>1

Find that g(u, v) assumes values {k, k + d, k + 2d, . . . , k + (q - 1)d}. Therefore, the labels of the edges of T constitute the set {k, k + d, k + 2d, . . . , k + (q - 1)d} and hence the mapping f is a (k, d) graceful labeling of G, i.e. G is a (k, d) graceful graph when d  k.

Theorem 3.3: The 3-distance tree derived from the 3-distance tree in Theorem 3.1 with some or all vertices of Pnattached to some leaves is (k, d) graceful with d  k.

Proof: Let T be a tree derived from the 3-distance tree in Theorem 3.1 with some or all the vertices of Pn may be adjacent to some leaves. Let the vertices on the central path Pn be c1,cn, ,cn. Let ci be attached to li leaves apart from the vertex ci,i. Let the li leaves adjacent to ci be xi,pi, pi = 1, 2, . . . , li. The remaining descriptions and notations involving the tree T are the same as those in Theorem 3.2. Define the mapping f : V (T ) - {0, 1, 2, 3, 4, , k + (q - 1)d} as follows.

For i = 1,2, , n, pi=1, 2, , li,  ji=1, 2, , mi, ti,ji=1, 2, , si,ji,

fci=i-1r+22d                    if i is oddk+q-i2r+2d    if i is even; fci,i=k+q-i-1r+22-1d   if i is oddi2r+2-1d                      if i is even;

fxi,pi=k+p1-1difi=1k+i-1r+22+p=1i-1lp+mp-r+jp=1rsp,z-li+r-ji+1difi>1 is oddq-ir+22-p=1i-1lp+mp-r+jp=1rsp,z-mi+r-z=1jisi,z-li-pi+1difi is even fci, i, ji=jidifjirq+r-li-jidifji>rifi=1i-1r+22+jidifjirq-i-1r+22-p=1i-1lp+mp-r+jp=1rsp, jp-li+r-ji+1difji>rifi>1 is oddk+q-i-2r+22-1-jidifjirk+ir+22+p=1i-1lp+mp-r+jp=1rsp, jp+ji-2difji>rifi is even fci,i, ji, ti, ji= k+l1+m1-r+t1,1difji=1k+l1+m1-r+z=1j1-1s1,z+j1-1+t1,j1difji>1ifi=1k+i-1r+22+p=1i-1lp+mp-r+z=1jpsp,z+li+mi-r+ti,1difji=1k+i-1r+22+ji+p=1i-1lp+mp-r+z=1jpsp,z+li+mi-r+z=1ji-1si,z+ti,ji-1difji>1ifi>1 is odd q-i-2r+22-p=1i-1lp+mp-r+z=1jpsp,z-li-mi+r-ti,1-1difji=1q-i-2r+22-ji-p=1i-1lp+mp-r+z=1jpsp,z-li-mi+r-z=1ji-1si,z-ti,jidifji>1ifi is even

Defining the labeling g  on E(T ) by g(u, v)  =  |f (u) - f (v)|, we have

for i  =  1, 2, . . . , n-1,  gci, ci+1=  k + q - ir + 2d;

for i  =  1, 2, . . . , n, ji  =  1, 2, . . . , mi, ti, i, ji=  1, 2, . . . , si, i, ji,

gci,ci,i=k+q-i-1r+2-1d   if i is oddk+q-ir+2-1d                  if i is even;

gci,xi,pi=k+p1-1difi=1k+p=1i-1lp+mp-r+jp=1rsp,z-li+r-ji+1dif i>1 is odd k+p=1i-1lp+mp-r+jp=1rsp,z+mi-r+z=1jisi,z+li+pi-1ifi is even; gci,i,ci,i,ji =k+q-j1-1difjirk+j1-r-1difji>rifi=1k+q-i-1r+2-ji-1difjirk+p=1i-1mp-r+jp=1rsp, jp-r+ji-1difji>rifi  is odd and i>1k+q-i-1r+2-jidifjirk+p=1i-1mp-r+jp=1rsp, jp-r+ji-1difji>rifi is even;

gci,i,ji, ci,i,ji,ti, i, ji

=k+m1-r+t1,1-1difji=1k+m1-r+z=1j1-1s1,z+t1,j1-1difji>1ifi=1k+p=1i-1mp-r+z=1jpsp, z+mi-r+ti,1-1difji=1k+p=1i-1mp-r+z=1jpsp, z+mi-r+z=1ji-1s1, z+ti, ji-1difji>1ifi>1  is odd.

Find that g(u, v) assumes values {k, k + d, k + 2d, . . . , k + (q - 1)d}. Therefore, the labels of the edges of T constitute the set {k, k + d, k + 2 d, . . . , k + (q - 1)d} and hence the mapping f is a (k, d) graceful labeling of T , i.e. T is a  (k, d) graceful tree with d  k.

Theorem 3.4 The three distant unicyclic graphs obtained from the three distant trees in Theorem 3.4 by joining the vertices c1 and cn admit (k, d) graceful labeling with d  k if n  (0 mod 4).

Proof: Consider the three distant unicyclic graph G in Theorem 3.4. Define the mapping

f : V (G) - {0, 1, 2, 3, 4, , k + (q - 1)d} as follows.

For i = 1,2, , n, pi=1, 2, , li,  ji=1, 2, , mi, ti,ji=1, 2, , si,ji,

fci= i-1r+22difi is odd and i<n2i-1r+22+1difi is odd and i>n2k+q-i2r+2difi is even

fci,i= k+q-12i-1r+2-1ifi is odd ir+22-1difi is even and in2i2r+2difi is even and i>n2;

fxi,pi=k+p1-1difi=1k+i-1r+22+p=1i-1lp+mp-r+z=1jpsp,z+pi-1dif1<i<n2, i oddk+i-1r+22+p=1i-1lp+mp-r+z=1jpsp,z+pidifi is odd, i>n2q-ir+22-p=1i-1lp+mp-r+z=1jpsp,z-mi+r+z=1jisi,z-pi+1difi is even; fci,i, ji= j1difjirq+r-j1difji>rifi=1i-1r+22+jidifjirq-i-1r+22-p=1i-1lp+mp-r+jp=1rsp, jp-li+r-jiifji>rifi is odd, 1 <i <n2i-1r+22+ji+1difjirq-i-1r+22-p=1i-1lp+mp-r+jp=1rsp, jp-li+r-jiifji>rifi is odd, i>n2k+q-i-2r+22-ji-1ifjirk+ir+22-p=1i-1lp+mp-r+jp=1rsp, jp+ji-2ifji>rifi is even, 1 <i n2k+q-i-2r+22-ji-1ifjirk+ir+22-p=1i-1lp+mp-r+jp=1rsp, jp+ji-1ifji>rifi is even, i>n2 fci,i, ji,ti,ji=  k+l1+m1-r+t1,1difji=1k+l1+m1-r+z=1j1-1s1,z+j1-1+t1,j1difji>1ifi=1k+i-1r+22+p=1i-1lp+mp-r+z=1jpsp,z+li+ mi-r+ti,1difji=1k+i-1r+22+ji+p=1i-1lp+mp-r+z=1jpsp,z+li+mi-r+z=1ji-1si,z+ti,ji-1difji>1ifi is odd, 1 <i <n2k+i-1r+22+p=1i-1lp+mp-r+z=1jpsp,z+li+mi-r+ti,1+1difji=1k+i-1r+22+ji+p=1i-1lp+mp-r+z=1jpsp,z+li+mi-r+z=1ji-1si,z+ti,jidifji>1ifi is odd and i>n2q-i-2r+22-ji-p=1i-1lp+mp-r+z=1jpsp,z-li-mi+r-ti,1-1difji=1q-i-2r+22-ji-p=1i-1lp+mp-r+z=1jpsp,z-li-mi+r-z=1ji-1si,z-ti,jidifji>1ifi is even .

Defining the labeling g on E(T ) by g(u, v)  =  |f (u) - f (v)|, we have

for i=1,2,,n-1, gci, ci+1=k+q-ir+2difi<n2k+q-ir+2-1difin2;

for i=1,2,,n,ji=1,2,,mi,ti,i, ji=1,2,, si,i, ji

gci,ci,i=k+q-i-1r+2-1difi is odd and i<n2k+q-i-1r+2-2difi is odd and i>n2k+q-ir+2-1difi is even and in2k+q-ir+2difi is even and i>n2; gci,xi,pi=k+p1-1difi=1k+p=1i-1lp+mp-r+z=1jpsp,z+pi-1dif i is odd and i>1k+p=1i-1lp+mp-r+z=1jpsp,z+mi-r-z=1jisi,z+pi-1ifi is even

k+q-j1-1difjirk+j1-r-1difji>rifi=1k+q-i-1r+2-j1-1difjirk+p=1i-1mp-r+jp=1rsp,jp-r+ji-1difji>rifi is odd, i<n2k+q-i-1r+2-j1-2difjirk+p=1i-1mp-r+jp=1rsp,jp-r+ji-1difji>rifi is odd, i>n2k+q-i-1r+2-j1difjirk+p=1i-1mp-r+jp=1rsp,jp-r+ji-1difji>rifi is even, in2k+q-i-1r+2-j1-1difjirk+p=1i-1mp-r+jp=1rsp,jp-r+ji-1difji>rifi is even, i>n2

gci,i,ji,ci,i,ji,ti,ji= k+m1-r+t1,1 -1difjirk+m1-r+z=1j1-1s1,z+t1,j1-1difji>rifi=1k+p=1i-1mp-r+z=1jpsp,z+mi-r+ti,1-1difjirk+p=1i-1mp-r+z=1jpsp,z+mi-r+z=1ji-1s1,z+ti,ji-1difji>rifi>1

Find that g(u, v) assumes values {k, k + d, k + 2d, . . . , k + (q - 1)d}. Therefore, the labels of the edges of T constitute the set {k, k + d, k + 2d, . . . , k + (q - 1)d} and hence the mapping f is a (k, d) graceful labeling of G, i.e. G is a (k, d) graceful graph when d  k.

Discussion: Figure 2, Figure 3, Figure 4 illustrate our results, i.e., Theorems 3.1 – 3.4

A three distance tree of the type in Theorem 3.1 with a graceful labeling. Here <inline-formula id="if-abaf1c2470df"> <tex-math>\begin{aligned} &n=8, m_{1}=5, m_{2}=5, m_{3}=5, m_{4}=5, m_{5}=5, m_{6}=5, m_{7}=5, m_{8}=5, r= \\ &3, s_{1,1}=5, s_{4,1}=1, s_{5,1}=1, s_{5,2}=1, s_{6,1}=1, s_{6,2}=5, s_{7,1}=4, s_{7,2}=2, s_{8,1}= \\ &1, s_{8,2}=5, s_{8,3}=1 \end{aligned}</tex-math></inline-formula>

<bold id="strong-b3996bd10b3d40a88b8640339d75860a"/>A three distance unicyclic graph of the type in Theorem 3.2 with a <inline-formula id="inline-formula-b6c6fd34fd204c85b66ebc989a38e50a"> <tex-math>(k,\;d)\;</tex-math></inline-formula>graceful labeling. Here <inline-formula id="inline-formula-5e44b6a8e92543ae834493fa18a89097"> <tex-math>n=8, m_{1}=5, m_{2}=3, m_{3}=4, m_{4}=5, m_{5}=5, m_{6}=5, m_{7}= 5, m_{8}=5, r=3, s_{1,1}=5, s_{2,1}=2, s_{3,1}=1 s_{4,1}=1, s_{5,1}=1, s_{5,2}=1, s_{6,1}= 1, s_{6,2}=5, s_{7,1}=4, s_{7,2}=2, s_{8,1}=1, s_{8,2}=5, s_{8,3}=1</tex-math></inline-formula> <bold id="strong-2dccb1bd70c64546b81de05a943f0a10"/><bold id="strong-b28b6741ba96411fb0210e55dd835eed"/>A three distance tree of the type in Theorem 3.3 with a <inline-formula id="inline-formula-ea00dcc1ae9a427a8eb946732ac04561"> <tex-math>(k,\;d)</tex-math></inline-formula> graceful labeling. Here <inline-formula id="inline-formula-4fcd4bcf148d491c9cfa793bec27d68a"> <tex-math>n\;=\;8,\;m_1=\;5,\;m_2\;=\;3,\;m_3=\;3,\;m_4=\;4,\;m_5=\;3,\;m_6\;=\;4,\;m_7=\;4,\;m_8=\;5,\;r\;=\;3,\;s_{1,1}=\;5,{\;s}_{3,1}=\;2,{\;s}_{3,2}=\;1,{\;s}_{3,3}=\;1,{\;s}_{5,1}=\;1,\;s_{5,2}=\;1,{\;s}_{6,1}=\;\;4,\;s_{6,2}=\;1,{\;s}_{7,1}=\;2,\;s_{7,2}=\;4,{\;s}_{8,1}=\;3,{\;s}_{8,2}=\;3,\;s_{8,3}=\;1,{\;l}_1=\;1,{\;l}_2=\;1,\;l_3=\;2,\;l_4=\;0,\;l_5=\;1,{\;l}_6=\;2,{\;l}_7=\;2,\;l_8=\;0.</tex-math></inline-formula> <bold id="strong-daa53e015d1440ca96b900e4f647564f"/><bold id="strong-78a6680adc004d66a8ec5c093c979b8c"/>A three distance unicyclic graph of the type in Theorem 3.4 with a <inline-formula id="inline-formula-3738ad9155ff49fe86e7339827cccb57"> <tex-math>(k,\;d)</tex-math></inline-formula> graceful labeling. Here <inline-formula id="inline-formula-3f6c689a068e41f697b6c76548b41313"> <tex-math>n\;=\;8,\;m_1=\;5,\;m_2\;=\;3,\;m_3=\;3,\;m_4=\;4,\;m_5=\;5,\;m_6\;=\;4,\;m_7=\;4,\;m_8=\;4,\;r\;=\;3,\;s_{1,1}=\;4,{\;s_{2,1}=\;1,s}_{3,1}=\;1,{\;s}_{5,1}=\;1,\;s_{5,2}=\;1,{\;s}_{6,1}=\;\;5,\;s_{6,2}=\;1,{\;s}_{7,1}=\;4,\;s_{7,2}=\;2,{\;s}_{8,1}=\;3,{\;s}_{8,2}=\;2,\;s_{8,3}=\;4,{\;l}_1=\;2,{\;l}_2=\;1,\;l_3=\;2,\;l_4=\;0,\;l_5=\;0,{\;l}_6=\;3,{\;l}_7=\;2,\;l_8=\;0.</tex-math></inline-formula> Conclusion

In this article we give \begin{pmatrix}k&d\end{pmatrix} graceful labeling to a class three distance trees which are obained from a firecracker by attaching leaves either to the leaves of the firecracker or the vertices on the central path of the firecracker or both. Find that in Construction 3.1 we assume the condition that m_{i} \geq r for each i Moreover, here we give \begin{pmatrix}k&d\end{pmatrix} graceful labeling to a class of three distance unicyclic graphs obtained by joining end vertices of the central path of a \begin{pmatrix}k&d\end{pmatrix} graceful three distance tree mentioned above. Our effort is the first of its kind where we give \begin{pmatrix}k&d\end{pmatrix}graceful labeling to a family of three distance three and three distance unicyclic graph. However, the future advancement of this result requires to cover all three distance trees and three distance unicyclic graphs, i.e., by generalizing our results dropping the assumption m_{i\;\geq\;r} for each i_{}

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