Pentagonal Graceful Labeling of Caterpillar Graphs

1M.P. Syed Ali Nisaya, 2D.S.T. Ramesh 1Assistant Professor, 2Associate Professor Department of Mathematics 1 The M.D.T. Hindu College, Tirunelveli – 627002, 2 Margoschis College, Nazareth- 628617, Tamil Nadu, India

Abstract - A graph G = (V, E) with p vertices and q edges is said to admit pentagonal graceful labeling if its vertices can be labeled by non negative integers such that the induced edge labels obtained by the absolute difference of the labels of end vertices are the first 풒 pentagonal numbers. A graph 푮 which admits pentagonal graceful labeling is called a pentagonal graceful graph. In this paper, we prove that Caterpillar is a pentagonal graceful graph.

Keywords - Pentagonal number, pentagonal graceful labeling, pentagonal graceful graph. ______

1. INTRODUCTION AND DEFINITIONS The graph considered here are finite, connected, undirected and simple. The vertex set and the edge set of a graph 퐺 are denoted by 푉(퐺) and 퐸(퐺) respectively. Terms not defined here are used in the sense of Harary [2]. For number theoretic terminology, [1] is followed. The definitions are useful for further investigations. A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. If the domain of the mapping is the set of vertices (edges / both) then the labeling is called a vertex (edge / total) labeling. There are several types of graph labeling and a detailed survey is found in [3]. The following definitions are necessary for present study. DEFINITION 1.1 Pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally

th symmetrical. The n pentagonal number, An is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. is given by the n(3n− 1) formula for n1 . The first few pentagonal numbers are 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 2 287, 330,… DEFINITION 1.2 Let G be a (푝, 푞) graph. A one to one function 푓: 푉(퐺) → {0,1,2, … , 푞} is called a graceful labeling of 퐺 if the induced edge labeling 푓′: 퐸(퐺) → {1,2, … , 푞} defined by 푓′(푒) = |푓(푢) − 푓(푣)| for each edge 푒 = 푢푣 of 퐺 is also one to one. A graph 퐺 possessing graceful labeling is called graceful graph. DEFINITION 1.3 [4] Let G be a (p, q) graph. Let V(G), E(G) denote the vertex set and the edge set of G respectively. th Consider an injective function where Aq is the q pentagonal number. That is, A1 = 1, A23== 5, A 12 etc.,

n(3n− 1) * * A = . Define the function f : E(G)→ {1,2,...,A } such that f (uv)=− f (u) f (v) for all edges n 2 q * uv E(G) . If f( E(G)) is a sequence of distinct consecutive pentagonal numbers say {A1 ,A 2 ,...,A q }then the function f is said to be pentagonal graceful labeling and the graph which admits such a labeling is called a pentagonal graceful graph.

2. MAIN RESULTS

Next we give some results on Caterpillars.

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2.1 Definition: Let v1 , v 2 , ..., v m be the m vertices of the path Pm. From each vertex v,i i = 1,2,…,m, there are n ,i= 1,2,...,m, pendent vertices say v ,v ,...,v . The resultant graph is a caterpillar and is denoted as i i1 i2 ini B(n ,n ,...,n ). 1 2 m

The graph B(n12 ,n ) is called a bistar graph. The caterpillar graph can also be defined in the following way. G is called a caterpillar if G is a tree such that the removal of the vertices with degree 1 results in a path. And that path is called the spine of the caterpillar. First, we prove three lemmas and then we prove a theorem on caterpillars.

Lemma1: The bistar B(n12 ,n ), where n1 ≥1 and n 2 ≥1 is pentagonal graceful. Proof: Let P be a path on two vertices and let v and v be the vertices of . From v,there are pendent vertices 2 1 2 1 say v ,v ,...,v and from v,there are n pendent vertices say v ,v ,...,v . The resulting graph is a 11 12 1n1 2 2 21 22 2n2 bistar B(n ,n ). 12 Let G = (V, E) be the bistar

Let V(G)= {v:ii = 1,2}  {v 1j :1  j  n} 1  {v 2 j :1  j  n} 2 and E(G)= {v v }  {v v :1  j  n }  {v v :1  j  n }. 1 2 1 1j 1 2 2 j 2 Then G has n++ n 1 edges. Let n+ n + 1 = m(say). 12 12

Now label the vertices v12 ,v of as 0 and 1.

Then label the n vertices adjacent to other than v as A ,A ,A ,...,A Next label the n 1 2 m m1− m2 − m(n1). −1 − 2 vertices adjacent to v2 other than as Am− n++ 1,...,A m − (n + n − 1) 1. 1 1 2 We shall prove that G admits pentagonal graceful labeling.

From the definition, it is clear that max f(v) is Am an f(v) {0,1,2,...,Am }. Also from the definition, all the v V(G) vertices of G have different labeling. Hence f is one to one.

It remains to show that the edge values are of the form{A1 ,A 2 ,...,A m }. The induced edge function * f :E(G)→ {1,2,...,Am } is defined as follows.

Am−− (j 1) if i= 1 and 1  j  n 1 *  f (vi v ij ) = 

A if i= 2 and 1  j  n  m− (n1 + j − 1) 2 * and f (v1 v 2 )= A 1 * * Clearly f is one to one and f (E(G))= {A1 ,A 2 ,...,A m }. Therefore G admits pentagonal graceful labeling.

Hence the graph B(n12 ,n ) is pentagonal graceful

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Illustration: The pentagonal graceful labeling of B(5,4) is shown in figure 1.

Figure 1

Lemma 2: The caterpillar B(n12 ,0,n ) is pentagonal graceful for all n12 ,n 1.

Proof: Let v ,v ,v be the three vertices of P . From v,there are n pendent vertices say u ,u ,...,u and 1 2 3 3 1 1 1 2 n1 from v,there are n pendent vertices say w ,w ,...,w . The resulting graph is denoted as . Let it be 3 2 1 2 n2

G= (V, E) . Then G has n12++ n 3 vertices and n12++ n 2 edges. Let n12+ n + 2 = m (say).

Define f :V(G)→ {0,1,2,...,Am }as follows. f (v1m )= A f (v2 )= 0 f (v )= A 3 m−− n1 1 f(u )= A − A , where 1  i  n i m m− i 1 f(w )= A + A , where 1  j  n . j m−− n1 1 j 2

We shall prove that G admits pentagonal graceful labeling. From the definition, it is clear that max f(v) is Am and v V(G)

Also from the definition, all the vertices of G have different labeling. Hence f is one to one. f(v) {0,1,2,...,Am }. It remains to show that the edge values are of the form {A ,A ,...,A }. The induced edge function 1 2 m * f :E(G)→ {1,2,..., A}m is defined as follows. * f (v1 v 2 )= A m * f (v v )= A 2 3 m−− n1 1 * f (v1 u i )= A m− i , where 1  i  n 1 * f (v3 w j )= A j , where 1  j  n 2

* * Clearly f is one to one and f (E(G))= {A1 ,A 2 ,...,A m } . Therefore G admits pentagonal graceful labeling.

Hence the graph B(n12 ,0,n ) is pentagonal graceful for all Illustration: The pentagonal graceful labeling of B(5,0,4) is given in figure 2.

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Figure 2

Lemma 3: The caterpillar B(n12 ,1,n ) is pentagonal graceful for all n12 ,n 1 . Illustration: The pentagonal graceful labeling of B(5,1,4) is given in figure 3.

Figure 3

2.2 Theorem: Caterpillars are pentagonal graceful.

Proof: Let v1 , v 2 , ..., v m be the m vertices of the path Pm. From each vertex v,i i = 1,2,…,m, there are n ,i= 1,2,...,m, pendent vertices say v ,v ,...,v . The resultant graph is a caterpillar and is denoted as i i1 i2 ini B(n ,n ,...,n ). Assume m 3. 1 2 m

Clearly B(n1 ,n 2 ,...,n m ) has n1+ n 2 + ... + n m + (m − 1) edges. Let n= n + n + ... + n + (m − 1). 1 2 m

Define f : V B(n ,n ,...,n )→ {0,1,2,...,A }as follows. ( 1 2 m) n f (v1 )= 0 f (v )== A , where i 1,2,...,n 1i n−− (i 1) 1 f (v )=+ f (v ) A 2 1 n− n1 f (v )= f (v ) − A , where i = 1,2,...,n 2i 2 n−− n1 i 2 f(v )=− f(v ) A 3 2 n− n12 − n − 1 f(v3i )= f(v 3 ) + A n− n − n − 1 − i , where i = 1,2,...,n 3 12 f(v4 )=+ f(v 3 ) A n− n − n − n − 2 1 2 3 f(v4i )= f(v 4 ) − A n− n − n − n − 2 − i , where i = 1,2,...,n 4 1 2 3

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© IJEDR 2018 | Volume 6, Issue 4 | ISSN: 2321-9939 and so on. f (v )− A if m is odd  m− 1 n − n1 − n 2 − − n m− 1 − (m − 2) f (vm ) =  f (v )+ A if m is even  m− 1 n − n1 − n 2 − − n m− 1 − (m − 2) f (v )− A if m is even and 1  i  n  mnnnn(m2)i−1 − 2 − − m− 1 − − − m f (vmi ) =  f (v )+ A if m is odd and 1  i  n  mnnnn(m2)i−1 − 2 − − m− 1 − − − m

Fori= n,f(v ) = f(v)  A m mnm m nnn− 1 − 2 − − n m− 1 − (m2)n − − m

=f(vm ) A n− n − n − − n − n − m + 2 1 2 m− 1 m =f (v ) A m1 Clearly the vertex labels are distinct and the resulting edge labels are of the form{A ,A ,...,A }. Thus caterpillars 1 2 m are pentagonal graceful. 2.3 Example: The pentagonal graceful labeling of the graph B(3,2,3,5) is given in figure 4.

Here m= 4,n1 = 3,n 2 = 2,n 3 = 3,n 4 = 5 . Therefore, n = 3 + 2 + 3 + 5 + 3 = 16 edges.

Figure 4 2.4 Conjecture: All trees are pentagonal graceful.

REFERENCE

[1] M. Apostal, Introduction to Analytic Number Theory, Narosa Publishing House, Second Edition – (1991) [2] Frank Harary, Graph Theory, Narosa Publishing House – (2001) [3] J.A. Gallian, A dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics, 16(2013), #DS6 [4] D.S.T. Ramesh and M.P. Syed Ali Nisaya, Some Important Results on Pentagonal Graceful Graphs, International Journal of Applied Mathematical Sciences(0973-0176), Volume 7, No.1, January – June (2014), 71-77

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