Divisor Cordial Labeling in the Context of Graph Operations …Divisor Cordial Labeling in the Context of Graph Operations on Bistar 2607 Deﬁnition 1.7. For a graph G the splitting

Global Journal of Pure and Applied Mathematics.ISSN 0973-1768 Volume 12, Number 3 (2016), pp. 2605–2618© Research India Publicationshttp://www.ripublication.com/gjpam.htm

Divisor Cordial Labeling in the Context of GraphOperations on Bistar

M. I. Bosmia

Government Engineering College,Sector-28, Gandhinagar, Gujarat, INDIA.

K. K. Kanani

Government Engineering College,Rajkot, Gujarat, INDIA.

Abstract

A divisor cordial labeling of a graph G with vertex set V (G) is a bijection f fromV (G) to {1, 2, . . . , |V (G)|} such that an edge e = uv is assigned the label 1 iff (u)|f (v) or f (v)|f (u) and the label 0 otherwise, then |ef (0) − ef (1)| ≤ 1. Agraph which admits a divisor cordial labeling is called a divisor cordial graph. Inthis paper we prove that bistar Bm,n, splitting graph of bistar Bm,n, degree splittinggraph of bistar Bm,n, shadow graph of bistar Bm,n, restricted square graph of bistarBm,n, barycentric subdivision of bistar Bm,n and corona product of bistar Bm,n withK1 admit divisor cordial labeling.

AMS subject classification: 05C78.Keywords: Divisor Cordial Labeling, Bistar.

1. Introduction

Throughout this work, by a graph we mean finite, connected, undirected, simple graphG = (V (G), E(G)) of order |V (G)| and size |E(G)|. For any undefined notationsand terminology we follow Gross and Yellen [4] while for number theory we followBurton [1].

Definition 1.1. If the vertices are assigned values subject to certain condition(s) then itis known as graph labeling.A dynamic survey on various graph labeling technique is regularly updated by Gallian [3].

2606 M. I. Bosmia and K. K. Kanani

Definition 1.2. A mapping f : V (G) → {0, 1} is called binary vertex labeling of G andf (v) is called the label of the vertex v of G under f .

Notation 1.3. If for an edge e = uv, the induced edge labeling f ∗ : E(G) → {0, 1} isgiven by f ∗(e = uv) = |f (u) − f (v)|. Then

vf (i) = number of vertices of G having label i under f ,ef (i) = number of edges of G having label i under f ∗.

Definition 1.4. A binary vertex labeling f of a graph G is called a cordial labeling if|vf (0) − vf (1)| ≤ 1 and |ef (0) − ef (1)| ≤ 1. A graph which admits cordial labeling iscalled a cordial graph.The cordial labeling technique was introduced by Cahit[2] in which he investigatedseveral results on this concept. After this many labeling techniques are also introducedwith minor changes in cordial labeling. The product cordial labeling, total product cordiallabeling and prime cordial labeling are some of them. The present work is focused ondivisor cordial labeling.

Definition 1.5. Let G = (V (G), E(G)) be a simple graph and f : V (G) → {1, 2, . . . ,

|V (G)|} be a bijection. For each edge e = uv, assign the label 1 if f (u)|f (v) orf (v)|f (u) and the label 0 otherwise. The function f is called a divisor cordial labelingif |ef (0) − ef (1)| ≤ 1. A graph which admits divisor cordial labeling is called a divisorcordial graph.The divisor cordial labeling technique was introduced by Varatharajan et al.[7] and theyproved the following results:

• The star graph K1,n is divisor cordial.

• The complete bipartite graph K2,n is divisor cordial.

• The complete bipartite graph K3,n is divisor cordial.

• S(K1,n), the subdivision of the star K1,n is divisor cordial.

Vaidya and Shah [6] proved that

• S′(Bn,n) is a divisor cordial graph.

• DS(Bn,n) is a divisor cordial graph.

• D2(Bn,n) is a divisor cordial graph.

• Restricted B2n,n is a divisor cordial graph.

Definition 1.6. Bistar Bm,n is the graph obtained by joining the center(apex) vertices ofK1,m and K1,n by an edge.

Divisor Cordial Labeling in the Context of Graph Operations on Bistar 2607

Definition 1.7. For a graph G the splitting graph S′(G) of a graph G is obtained byadding a new vertex v′ corresponding to each vertex v of G such that N(v) = N(v′).

Definition 1.8. [5] Let G = (V (G), E(G)) be a graph with V = S1 ∪ S2 ∪ · · · ∪ St ∪T

where each Si is a set of vertices having at least two vertices of the same degree and

T = V −( t⋃

i=1

Si

). The degree splitting graph of G denoted by DS(G) is obtained from

G by adding vertices w1, w2, w3, . . . , wt and joining to each vertex of Si for 1 ≤ i ≤ t .

Definition 1.9. The shadow graph D2(G) of a connected graph G is constructed bytaking two copies of G say G′ and G". Join each vertex u′ in G′ to the neighbours of thecorresponding vertex u" in G".

Definition 1.10. For a simple connected graph G the square of graph G is denoted by G2

and defined as the graph with the same vertex set as of G and two vertices are adjacentin G2 if they are at a distance 1 or 2 apart in G.We note that the restricted square graph of bistar B2

m,n is the graph obtained from Bm,n

by joining all the pendant vertices of the K1,m with the apex vertex of K1,n and all thependant vertices of the K1,n with the apex vertex of K1,m.

Definition 1.11. Let G = (V (G), E(G)) be a graph. Let e = uv be an edge of G and w

is not a vertex of G. The edge e is sub divided when it is replaced by the edges e′ = uw

and e′′ = wv.

Definition 1.12. Let G = (V (G), E(G)) be a graph. If every edge of graph G issubdivided, then the resulting graph is called barycentric subdivision of graph G. Inother words barycentric subdivision is the graph obtained by inserting a vertex of degree2 into every edge of original graph. The barycentric subdivision of any graph G isdenoted by S(G).

Definition 1.13. If G is graph of order n, the corona of G with another graph H , G�H

is the graph obtained by taking one copy of G and n copies of H and joining the ith

vertex of G with an edge to every vertex in the ith copy of H .

2. Main Results

Theorem 2.1. The bistar Bm,n is a divisor cordial graph.

Proof. Let G = Bm,n be the bistar with vertex set V (G) = {u0, v0, ui, vj : 1 ≤ i ≤m, 1 ≤ j ≤ n}, where ui and vj are pendant vertices. We note that |V (G)| = m+n+ 2and |E(G)| = m+n+ 1. Without loss of generality we can assume that m ≤ n becauseBm,n and Bn,m are isomorphic graphs.

Define vertex labeling f : V (G) → {1, 2, . . . , m + n + 2} as follows:f (u0) = 1, f (v0) = 2.


f (ui) = 2i + 2; 1 ≤ i ≤ m.

f (vj ) =

2j + 1; 1 ≤ j ≤⌈m + n

2

⌉

2m + 2 + 2

(j −

⌈m + n

2

⌉);

⌈m + n

2

⌉< j ≤ n

In view of the above defined labeling pattern we have ef (0) =⌊m + n + 1

2

⌋and ef (1) =⌈m + n + 1

2

⌉. Thus, |ef (0) − ef (1)| ≤ 1. Hence, the bistar Bm,n is a divisor cordial

graph. �

Illustration 2.2. Divisor cordial labeling of the graph B4,8 is shown in the Figure 1.

Figure 1: Divisor cordial labeling of B4,8.








Theorem 2.6. S′(Bm,n) is a divisor cordial graph.

Proof. Let Bm,n be the bistar with vertex set V (Bm,n) = {u0, v0, ui, vj : 1 ≤ i ≤m, 1 ≤ j ≤ n}, where ui and vj are pendant vertices. Let u′

0, v′0, u

′i , v

′j be the newly

added vertices in order to obtain G = S′(Bm,n), where 1 ≤ i ≤ m and 1 ≤ j ≤ n. Wenote that |V (G)| = 2m+ 2n+ 4 and |E(G)| = 3m+ 3n+ 3. Without loss of generalitywe can assume that m ≤ n because S′(Bm,n) and S′(Bn,m) are isomorphic graphs.Define vertex labeling f : V (G) → {1, 2, . . . , 2m + 2n + 4} as follows:f (u0) = 1, f (u′

0) = 2m + 2n + 3.f (v0) = 2, f (v′

0) = 4.f (ui) = 2n + 1 + 2i; 1 ≤ i ≤ m.

f (u′i) = 4

(n + 1 −

⌈m + n

2

⌉)+ 4i; 1 ≤ i ≤ m.

f (vj ) =

4j + 2; 1 ≤ j ≤⌈m + n

2

⌉

4

(j −

⌈m + n

2

⌉)+ 4;

⌈m + n

2

⌉< j ≤ n


f (v′j ) = 2j + 1; 1 ≤ j ≤ n.

In view of the above defined labeling pattern we have ef (0) =⌊3m + 3n + 3

2

⌋and

ef (1) =⌈3m + 3n + 3

2

⌉. Thus, |ef (0) − ef (1)| ≤ 1. Hence, S′(Bm,n) is a divisor

cordial graph. �

Illustration 2.7. Divisor cordial labeling of the graph S′(B3,7) is shown in the Figure 5.

Figure 5: Divisor cordial labeling of S′(B3,7).






Theorem 2.10. DS(Bm,n) is a divisor cordial graph.

Proof. Let Bm,n be the bistar with vertex set V (Bm,n) = {u0, v0, ui, vj : 1 ≤ i ≤m, 1 ≤ j ≤ n}, where ui and vj are pendant vertices. Here, V (Bm,n) = V1 ∪ V2, whereV1 = {ui, vj : 1 ≤ i ≤ m, 1 ≤ j ≤ n} and V2 = {u0, v0}. Without loss of generality wecan assume that m ≤ n because DS(Bm,n) and DS(Bn,m) are isomorphic graphs. Nowin order to obtain G = DS(Bm,n) from Bm,n, we consider following two cases.

Case 1: m = n.We add w1, w2 corresponding to V1, V2. Then |V (G)| = 2n+4 and E(G) = E(Bn,n)∪{u0w2, v0w2, uiw1, viw1 : 1 ≤ i ≤ n}. So, |E(G)| = 4n + 3.

Define vertex labeling f : V (G) → {1, 2, . . . , 2n + 4} as follows.Let p be the largest prime number such that p < 2n + 4.f (u0) = 2, f (v0) = 1.f (w1) = p, f (w2) = 3.f (ui) = 2i + 2; 1 ≤ i ≤ n.

Label the remaining verticesv1, v2, . . . , vn from the set {5, 7, 9, . . . , 2n+3, 2n+4}−{p}.In view of the above defined labeling pattern we have ef (0) = 2n + 1, ef (1) = 2n + 2.Thus, |ef (0) − ef (1)| ≤ 1.


Case 2: m < n.We add w1 to V1. Then |V (G)| = m + n + 3 and E(G) = E(Bm,n) ∪ {uiw1, vjw1 :1 ≤ i ≤ m, 1 ≤ j ≤ n}. So, |E(G)| = 2m + 2n + 1.

Define vertex labeling f : V (G) → {1, 2, . . . , m + n + 3} as follows:Let q be the largest prime number such that q ≤ m + n + 3.f (u0) = 2, f (v0) = 1.f (w1) = q.f (ui) = 2i + 2; 1 ≤ i ≤ m.

Label the remaining vertices v1, v2, . . . , vn from the set {3, 5, 7, . . . , 2m + 3, 2m +4, 2m + 5, . . . , m + n + 3} − {q}. In view of the above defined labeling pattern we haveef (0) = m + n, ef (1) = m + n + 1. Thus, |ef (0) − ef (1)| ≤ 1. Hence, DS(Bm,n) is adivisor cordial graph. �

Illustration 2.11. Divisor cordial labeling of the graph DS(B4,4) is shown in the Fig-ure 8.

Figure 8: Divisor cordial labeling of DS(B4,4).


Illustration 2.12. Divisor cordial labeling of the graph DS(B3,6) is shown in the Fig-ure 9.

Figure 9: Divisor cordial labeling of DS(B3,6).

Theorem 2.13. D2(Bm,n) is a divisor cordial graph.

Proof. Let G′ and G" be two copies of bistar Bm,n. Let V (G′) = {u′0, v

′0, u

′i , v

′j : 1 ≤

i ≤ m, 1 ≤ j ≤ n} and V (G") = {u"0, v"0, u"i , v"j : 1 ≤ i ≤ m, 1 ≤ j ≤ n}. LetG = D2(Bm,n). We note that |V (G)| = 2m + 2n + 4 and |E(G)| = 4m + 4n + 4.Without loss of generality we can assume that m ≤ n because D2(Bm,n) and D2(Bn,m)

are isomorphic graphs.

Define vertex labeling f : V (G) → {1, 2, . . . , 2m + 2n + 4} as follows:Let p be the largest prime number and q be the second largest prime number such thatq < p < 2m + 2n + 4.

f (u′0) = 2 , f (u"0) = q.

f (v′0) = 1 , f (v"0) = p.

f (u′i) = 2i + 2; 1 ≤ i ≤ m.

f (u"i) = 2m + 2 + 2i; 1 ≤ i ≤ m.

Label the remaining vertices v′1, v

′2, . . . , v

′n, v"1, v"2, . . . , v"n from the set {3, 5, 7, . . . ,

4m + 3, 4m + 4, 4m + 5, . . . , 2m + 2n + 4} − {p, q}. In view of the above definedlabeling pattern we have ef (0) = ef (1) = 2m + 2n + 2. Thus, |ef (0) − ef (1)| ≤ 1.Hence, D2(Bm,n) is a divisor cordial graph. �


Illustration 2.14. Divisor cordial labeling of the graph D2(B4,8) is shown in the Fig-ure 10.

Figure 10: Divisor cordial labeling of D2(B4,8).

Theorem 2.15. Restricted B2m,n is a divisor cordial graph.

Proof. Let Bm,n be the bistar with vertex set V (Bm,n) = {u0, v0, ui, vj : 1 ≤ i ≤ m, 1 ≤j ≤ n}, where ui and vj are pendant vertices. Let G be the restricted B2

m,n graph withV (G) = V (Bm,n) and E(G) = E(Bm,n) ∪ {v0ui, u0vj : 1 ≤ i ≤ m, 1 ≤ j ≤ n}. Wenote |V (G)| = m + n + 2 and |E(G)| = 2m + 2n + 1. Without loss of generality wecan assume that m ≤ n because restricted B2

m,n and restricted B2n,m are isomorphic graphs.

Define vertex labeling f : V (G) → {1, 2, . . . , m + n + 2} as follows:Let p be the largest prime number such that p ≤ m + n + 2.f (u0) = p, f (v0) = 1.f (ui) = 2i; 1 ≤ i ≤ m.

Label the remaining vertices v1, v2, . . . , vn from the set {3, 5, 7, . . . , 2m + 1, 2m +2, 2m + 3, . . . , m + n + 2} − {p}. In view of the above defined labeling pattern we haveef (0) = m + n and ef (1) = m + n + 1. Thus, |ef (0) − ef (1)| ≤ 1. Hence, restrictedB2

m,n is a divisor cordial graph. �






Theorem 2.18. The barycentric subdivisionS(Bm,n)of the bistarBm,n is a divisor cordialgraph.


Proof. Let Bm,n be the bistar with vertex set V (Bm,n) = {u0, v0, ui, vj : 1 ≤ i ≤ m, 1 ≤j ≤ n}, where ui and vj are pendant vertices and edge set E(Bm,n) = {u0v0, u0ui, v0vj :1 ≤ i ≤ m, 1 ≤ j ≤ n}. Let w0, w1, w2, . . . , wm, w′

1, w′2, . . . , w

′n be the newly added

vertices to obtain G = S(Bm,n). Where w0 is added between u0 and v0, wi is addedbetween u0 and ui for 1 ≤ i ≤ m and w′

j is added between v0 and vj for 1 ≤ j ≤ n.We note that |V (G)| = 2m + 2n + 3 and |E(G)| = 2m + 2n + 2.

Define vertex labeling f : V (G) → {1, 2, . . . , 2m + 2n + 3} as follows:f (u0) = 2, f (v0) = 1.f (w0) = 3.f (ui) = 2i + 3; 1 ≤ i ≤ m.f (wi) = 2i + 2; 1 ≤ i ≤ m.f (vj ) = 2m + 3 + 2j ; 1 ≤ j ≤ n.f (w′

j ) = 2m + 2 + 2j ; 1 ≤ j ≤ n.

In view of the above defined labeling pattern we have ef (0) = ef (1) = m+n+1. Thus,|ef (0) − ef (1)| ≤ 1. Hence, the barycentric subdivision S(Bm,n) of the bistar Bm,n is adivisor cordial graph. �

Illustration 2.19. Divisor cordial labeling of the graph S(B3,7) is shown in the Figure 13.

Figure 13: Divisor cordial labeling of S(B3,7).

Theorem 2.20. Bm,n � K1 is a divisor cordial graph.

Proof. Let Bm,n be the bistar with vertex set V (Bm,n) = {u0, v0, ui, vj : 1 ≤ i ≤ m, 1 ≤j ≤ n}, where ui and vj are pendant vertices. Let u′

0, v′0, u

′1, u

′2, . . . , u

′m, v′

1, v′2, . . . , v

′n

be the newly added vertices to obtain the graph G = Bm,n � K1. We note thatV (G) = V (Bm,n) ∪ {u′

0, v′0, u

′i , v

′j : 1 ≤ i ≤ m, 1 ≤ j ≤ n} and E(G) = E(Bm,n) ∪

{u0u′0, v0v

′0, uiu

′i , vjv

′j : 1 ≤ i ≤ m, 1 ≤ j ≤ n}. Hence, |V (G)| = 2m + 2n + 4 and

|E(G)| = 2m + 2n + 3.


Define vertex labeling f : V (G) → {1, 2, . . . , 2m + 2n + 4} as follows:f (u0) = 1 , f (u′

0) = 2m + 2n + 4.f (v0) = 2 , f (v′

0) = 3.f (ui) = 2i + 2; 1 ≤ i ≤ m.f (u′

i) = 2i + 1; 1 ≤ i ≤ m.f (vj ) = 2m + 2 + 2j ; 1 ≤ j ≤ n.f (v′

j ) = 2m + 3 + 2j ; 1 ≤ j ≤ n.

In view of the above defined labeling pattern we have ef (0) = m + n + 1 and ef (1) =m + n + 2. Thus, |ef (0) − ef (1)| ≤ 1. Hence, Bm,n � K1 is a divisor cordial graph.

�

Illustration 2.21. Divisor cordial labeling of the graph B5,7 � K1 is shown in the Fig-ure 14.

Figure 14: Divisor cordial labeling of B5,7 � K1.

3. Concluding Remark

To explore some new divisor cordial graphs is an open problem.

References

[1] D. M. Burton, Elementary Number Theory, Brown Publishers, Second Edition,(1990).

[2] I. Cahit, Cordial Graphs, A weaker version of graceful and harmonious graphs, ArsCombinatoria, 23(1987), 201–207.


[3] J. A. Gallian, A dynamic Survey of Graph labeling, The Electronics Journal ofCombinatorics, 18(2015), # D56.

[4] J. Gross and J. Yellen, Handbook of Graph Theory, CRC Press, (2004).

[5] P. Selvaraju, P. Balaganesan, J. Renuka and V. Balaj, Degree spliting graph on grace-ful, felicitous and elegant labeling, International journal of Mathematical combina-torics, 2(2012), 96–102.

[6] S. K. Vaidya, N. H. Shah, Some star and bistar related divisor cordial graphs, Annalsof Pure and Applied Mathematics, 3(1) (2013), 67–77.

[7] R.Varatharajan, S.Navanaeethakrishnan, K.Nagarajan, Divisor cordial graphs, In-ternational Journal of Mathematical Combinatorics, 4(2011), 15–25.

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Divisor Cordial Labeling in the Context of Graph Operations …Divisor Cordial Labeling in the Context of Graph Operations on Bistar 2607 Deﬁnition 1.7. For a graph G the splitting