A prime cordial labeling of a graph G with the vertex set V(G) is a bijection f:V(G)→{1,2,3,…,|V(G)|} such that each edge uv is assigned the label 1 if gcd(f(u),f(v))=1 and 0 if gcd(f(u),f(v))>1; then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph which admits a prime cordial labeling is called a prime cordial graph. In this work we give a method to construct larger prime cordial graph using a given prime cordial graph G. In addition to this we have investigated the prime cordial labeling for double fan and degree splitting graphs of path as well as bistar. Moreover we prove that the graph obtained by duplication of an edge (spoke as well as rim) in wheel Wn admits prime cordial labeling.
1. Introduction
We consider a finite, connected, undirected, and simple graph G=(V(G),E(G)) with p vertices and q edges which is also denoted as G(p,q). For standard terminology and notations related to graph theory we follow Balakrishnan and Ranganathan [1] while for any concept related to number theory we refer to Burton [2]. In this section we provide brief summary of definitions and other required information for our investigations.
Definition 1.
The Graph labeling is an assignment of numbers to the vertices or edges or both subject to certain condition(s). If the domain of the mapping is the set of vertices (edges), then the labeling is called a vertex labeling (edge labeling).
Many labeling schemes have been introduced so far and they are explored as well by many researchers. Graph labelings have enormous applications within mathematics as well as to several areas of computer science and communication networks. Various applications of graph labeling are reported in the work of Yegnanaryanan and Vaidhyanathan [3]. For a dynamic survey on various graph labeling problems along with an extensive bibliography we refer to Gallian [4].
Definition 2.
A labeling f:V(G)→{0,1} is called binary vertex labeling of G and f(v) is called the label of the vertex v of G under f.
Notation 1.
If for an edge e=uv, the induced edge labeling f*:E(G)→{0,1} is given by f*(e)=|f(u)-f(v)|. Then
(1)vf(i)=numberofverticesofGhavinglabeliunderfef(i)=numberofedgesofGhavinglabeliunderf*,jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjwherei=0or1.
Definition 3.
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 G is cordial if it admits cordial labeling.
The concept of cordial labeling was introduced by Cahit [5].
The notion of prime labeling was originated by Entringer and was introduced by Tout et al. [6].
Definition 4.
A prime labeling of a graph G is an injective function f:V(G)→{1,2,…,|V(G)|} such that for, every pair of adjacent vertices u and v, gcd(f(u),f(v))=1. The graph which admits a prime labeling is called a prime graph.
The concept of prime labeling has attracted many researchers as the study of prime numbers is of great importance because prime numbers are scattered and there are arbitrarily large gaps in the sequence of prime numbers. Vaidya and Prajapati [7, 8] have investigated many results on prime labeling. Same authors [9] have discussed prime labeling in the context of duplication of graph elements. Motivated through the concepts of prime labeling and cordial labeling, a new concept termed as a prime cordial labeling was introduced by Sundaram et al. [10] which contains blend of both the labelings.
Definition 5.
A prime cordial labeling of a graph G with vertex set V(G) is a bijection f:V(G)→{1,2,3,…,|V(G)|} and if the induced function f*:E(G)→{0,1} is defined by
(2)f*(e=uv)=1,ifgcd(f(u),f(v))=1,=0,otherwise,
then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph which admits prime cordial labeling is called a prime cordial graph.
Many graphs are proved to be prime cordial in the work of Sundaram et al. [10]. Prime cordial labeling for some cycle related graphs has been discussed by Vaidya and Vihol [11]. Prime cordial labeling in the context of some graph operations has been discussed by Vaidya and Vihol [12] and Vaidya and Shah [13, 14]. Vaidya and Shah [14] have proved that the wheel graph Wn admits prime cordial labeling for n≥8 while same authors in [15] have discussed prime cordial labeling for some wheel related graphs. Babitha and Baskar Babujee [16] have exhibited prime cordial labeling for some cycle related graphs and discussed the duality of prime cordial labeling. The same authors in [17] have derived some characterizations of prime cordial graphs and investigated various methods to construct larger prime cordial graphs using existing prime cordial graphs. We investigate a method different from existing one to construct larger prime cordial graph from an existing prime cordial graph.
Definition 6.
The wheel Wn is defined to be the join K1+Cn. The vertex corresponding to K1 is known as apex and vertices corresponding to cycle are known as rim vertices while the edges corresponding to cycle are known as rim edges.
Definition 7.
The bistar Bn,n is a graph obtained by joining the center (apex) vertices of two copies of K1,n by an edge.
Definition 8.
The fan Fn is the graph obtained by taking n-2 concurrent chords in cycle Cn+1. The vertex at which all the chords are concurrent is called the apex vertex. In other words, Fn=Pn+K1.
Definition 9.
The double fan DFn consists of two fan graphs that have a common path. In other words, DFn=Pn+K2____.
Definition 10.
The duplication of an edge e=uv of graph G produces a new graph G′ by adding an edge e′=u′v′ such that N(u′)=N(u)∪{v′}-{v} and N(v′)=N(v)∪{u′}-{u}.
Definition 11 (see [18]).
Let G=(V(G),E(G)) be a graph with V=S1∪S2∪S3∪⋯St∪T, where each Si is a set of vertices having at least two vertices of the same degree and T=V∖∪i=1tSi. 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.
2. Main ResultsTheorem 12.
Let G(p,q) with p≥4 be a prime cordial graph and let K2,n be a bipartite graph with bipartition V=V1∪V2 with V1={v1,v2} and V2={u1,u2,…,un}. If G1 is the graph obtained by identifying the vertices v1 and v2 of K2,n with the vertices of G having labels 2 and 4, respectively, then G1 admits prime cordial labeling in any of the following cases:
nis even and G is of any size q;
n,p,andq are odd with ef(0)=⌊q/2⌋;
n is odd, p is even, and q is odd with ef(0)=⌈q/2⌉.
Proof.
Let G(p,q) be a prime cordial graph and let f1 be the prime cordial labeling of G. Let w1,w2∈V be the vertices of G such that f1(w1)=2 and f1(w2)=4. Consider the K2,n with bipartition V=V1∪V2 with V1={v1,v2} and V2={u1,u2,…,un}. Now identify the vertices v1 to w1 and v2 to w2 and denote the resultant graph as G1. Then V(G1)=V(G)∪{u1,u2,…,un} and E(G1)=E(G)∪{w1ui,w2ui/1≤i≤n} so |V(G1)|=p+n and |E(G1)|=q+2n. To define f:V(G1)→{1,2,3,…,p+n}, we consider the following three cases.
Case (i) (n is even and G is of any size q). Since G is a prime cordial graph, we assign vertex labels such that f(wi)=f1(wi), where wi∈V(G)=V(G1)∩V(G) and i≤i≤p:
(3)f(ui)=p+i,i≤i≤n.
Since n is even and f1(w1)=2 and f1(w2)=4, w1 and w2 are adjacent to each ui, i≤i≤n. And this vertex assignment generates n edges with label 1 and n edges with label 0. Following Table 1 gives edge condition for prime cordial labeling for G1 under f.
From Table 1, we have |ef(0)-ef(1)|≤1.
Case (ii) (n,p,andq are odd with ef(0)=⌊q/2⌋). Here p and q both are odd and G is a prime cordial graph with ef(0)=⌊q/2⌋.
Since G is a prime cordial graph, we keep the vertex label of all the vertices of G in G1 as it is. Therefore f(wi)=f1(wi), where wi∈V(G) and i≤i≤p:
(4)f(ui)=p+i,i≤i≤n.
Since n is odd and f1(w1)=2 and f1(w2)=4, w1 and w2 are adjacent to each ui, i≤i≤n. And this vertex assignment generates n+1 edges with label 0 and n-1 edges with label 1.
Therefore edge conditions for G1 under f are ef(0)=⌊q/2⌋+n+1 and ef(1)=⌈q/2⌉+n-1. Therefore, ef(0)-1=ef(1). Hence, |ef(0)-ef(1)|≤1 for graph G1.
Case (iii) (n is odd, p is even, and q is odd with ef(0)=⌈q/2⌉). Here p is even, q is odd, and G is a prime cordial graph with ef(0)=⌈q/2⌉.
Since G is a prime cordial graph, we keep the vertex label of all the vertices of G in G1 as it is. Therefore f(wi)=f1(wi), where wi∈V(G) and i≤i≤p:
(5)f(ui)=p+i,i≤i≤n.
Since n is odd and f1(w1)=2 and f1(w2)=4, w1 and w2 are adjacent to each ui, i≤i≤n. And this vertex assignment generates n-1 edges with label 0 and n+1 edges with label 1.
Therefore edge conditions for G1 under f are ef(0)=⌈q/2⌉+n-1 and ef(1)=⌊q/2⌋+n+1. Therefore, ef(0)=ef(1)-1. Hence, |ef(0)-ef(1)|≤1 for graph G1.
Hence, in all the cases discussed above, G1 admits prime cordial labeling.
q
Edge conditions for G
Edge conditions for G1
Even
ef(0)=ef(1)=q2
ef(0)=ef(1)=q2
Odd
ef(0)-1=ef(1)=⌊q2⌋
ef(0)-1=ef(1)=⌊q2⌋+n
ef(0)=ef(1)-1=⌊q2⌋
ef(0)=ef(1)-1=⌊q2⌋+n
Illustration 1. Consider the graph G as shown in Figure 1, with p=7 and q=9. G is a prime cordial graph with ef(0)=4,ef(1)=5. Take n=3 and construct graph G1. In accordance with Case (ii) of Theorem 12, a prime cordial labeling of G1 is as shown in Figure 1. Here ef(0)=8,ef(1)=7.
Theorem 13.
Double fan DFn is a prime cordial graph for n=8 and n≥10.
Proof.
Let DFn be the double fan with apex vertices u1,u2 and v1,v2,…,vn are vertices common path. Then |V(DFn)|=n+2 and |E(DFn)|=3n-1. To define f:V(G)→{1,2,3,…,n+2}, we consider the following five cases.
Case 1 (n=3 to 7 and n=9). In order to satisfy the edge condition for prime cordial labeling in DF3 it is essential to label four edges with label 0 and four edges with label 1 out of eight edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most one edge and 1 label for at least seven edges. That is, |ef(0)-ef(1)|=6>1. Hence, DF3 is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in DF4 it is essential to label five edges with label 0 and six edges with label 1 out of eleven edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most three edges and 1 label for at least eight edges. That is, |ef(0)-ef(1)|=5>1. Hence, DF4 is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in DF5 it is essential to label seven edges with label 0 and seven edges with label 1 out of fourteen edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most four edges and 1 label for at least ten edges. That is, |ef(0)-ef(1)|=6>1. Hence, DF5 is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in DF6 it is essential to label eight edges with label 0 and nine edges with label 1 out of seventeen edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most six edges and 1 label for at least eleven edges. That is, |ef(0)-ef(1)|=5>1. Hence, DF6 is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in DF7 it is essential to label ten edges with label 0 and ten edges with label 1 out of twenty edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most eight edges and 1 label for at least twelve edges. That is, |ef(0)-ef(1)|=4>1. Hence, DF7 is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in DF9 it is essential to label thirteen edges with label 0 and thirteen edges with label 1 out of twenty-six edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most twelve edges and 1 label for at least fourteen edges. That is, |ef(0)-ef(1)|=2>1. Hence, DF9 is not a prime cordial graph.
Case 2 (n=8,10,11,12). For n=8, f(u1)=2, f(u2)=6 and f(v1)=4, f(v2)=8, f(v3)=3, f(v4)=9, f(v5)=10, f(v6)=5, f(v7)=7, and f(v8)=1. Then ef(0)=11, ef(1)=12.
For n=10, f(u1)=2, f(u2)=6 and f(v1)=3, f(v2)=9, f(v3)=12, f(v4)=8,f(v5)=4, f(v6)=10, f(v7)=1, f(v8)=5, f(v9)=7, and f(v10)=11. Then ef(0)=15, ef(1)=14.
For n=11, f(u1)=2, f(u2)=6 and f(v1)=3, f(v2)=9, f(v3)=12, f(v4)=8, f(v5)=4, f(v6)=10, f(v7)=5, f(v8)=7, f(v9)=11, f(v10)=13, and f(v11)=1. Then ef(0)=16, ef(1)=16.
For n=12, f(u1)=2, f(u2)=6 and f(v1)=3, f(v2)=9, f(v3)=12, f(v4)=8, f(v5)=4, f(v6)=14, f(v7)=10, f(v8)=5, f(v9)=7, f(v10)=11, f(v11)=13, and f(v12)=1. Then ef(0)=18, ef(1)=17.
Now for the remaining three cases let
(6)k=⌊n+22⌋,m=⌊n+23⌋,t1=⌈3n-12⌉,t2=3k-7+⌈m2⌉,t3= largest even number ≤n+2, and t4= largest odd number ≤n+2.
Case 3(t1=t2). Consider
(7)f(u1)=2,f(u2)=6.
For the vertices v1,v2,v3,…,vn we assign the vertex labels in the following order: 1, t3, t3-2, t3-4, …, 14, 12, 10, 8, 4, 3, 5, 7, 9, …, t4-2, t4.
Case 4(t1>t2). Consider
(8)f(u1)=2,f(u2)=6.
Let t5=t1-t2. Consider
(9)f(v1)=3,f(v2)=9,f(v3)=15,f(v4)=21,⋮f(vt5)=3(2t5-1),f(vt5+1)=f(vt5)+6.
Now for remaining vertices vt5+2,vt5+3,…,vn assign the labels 1,t3,t3-2,t3-4,…,14,12,10,8,4,5,7,…, all the odd numbers in ascending order.
Case5(t2>t1). Let t6=t2-t1.
Sub-Case 1. n is even. Consider
(10)f(u1)=2,f(u2)=6.
For the vertices v1,v2,v3,…,vn we assign the vertex labels in the following order: n+2, n+1, n, n-1, n-2, n-3,n+2-2t6, n+2-2(t6+1), n+2-2(t6+2),…,10,8,4,3,5,7,… remaining odd numbers in ascending order.
Sub-Case 2. n is odd. Consider
(11)f(u1)=2,f(u2)=6.
For the vertices v1,v2,v3,…,vn we assign the vertex labels in the following order: n+2, n+1, n, n-1, n-2, n-3, n+2-2t6, n+2-2(t6+1), n+2-2(t6+2),…,10,8,4,3,5,7,…remaining odd numbers in ascending order.
In view of the above defined labeling pattern for Cases 3, 4, and 5, we have
(12)ef(0)=⌈3n-12⌉,ef(1)=⌊3n-12⌋.
Thus, we have |ef(0)-ef(1)|≤1.
Hence, DFn is a prime cordial graph for n=8 and n≥10.
Illustration 2. For the graph DF15, |V(DF15)|=17 and |E(DF15)|=44. In accordance with Theorem 13 we have k=5, m=5, t1=22, and t2=20. Here t1>t2 so labeling pattern described in Case 4 will be applicable and t5=2. The corresponding prime cordial labeling is shown in Figure 2. Here ef(0)=22=ef(1).
Illustration 3. For the graph DF37, |V(DF37)|=39 and |E(DF37)|=110. In accordance with Theorem 13, we have k=19, m=13, t1=55, and t2=57. Here t2>t1 and n=37 so labeling pattern described in Sub-Case 2 of Case 5 will be applicable and t6=2. And corresponding labeling pattern is as below:
(13)f(u1)=2,f(u2)=6.
For the vertices v1,v2,…,v37 we assign the vertex labels 39, 38, 37, 36, 35, 34, 32, 30, 28, 26, 24, 22, 20, 18, 16, 14, 12, 10, 8, 4, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, and 33, respectively, where ef(0)=55=ef(1). Therefore DF37 is a prime cordial graph.
Theorem 14.
The graph obtained by duplication of an arbitrary rim edge by an edge in Wn is a prime cordial graph, where n≥6.
Proof.
Let v0 be the apex vertex of Wn and let v1,v2,…,vn be the rim vertices. Without loss of generality we duplicate the rim edge e=v1v2 by an edge e′=u1u2 and call the resultant graph as G. Then |V(G)|=n+3 and |E(G)|=2n+5. To define f:V(G)→{1,2,3,…,n+3}, we consider the following four cases.
Case 1(n=3,4,5). For n=3, to satisfy the edge condition for prime cordial labeling, it is essential to label five edges with label 0 and six edges with label 1 out of eleven edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most four edges and 1 label for at least seven edges. That is, |ef(0)-ef(1)|=3>1. Hence, for n=3, G is not a prime cordial graph.
For n=4, to satisfy the edge condition for prime cordial labeling, it is essential to label six edges with label 0 and seven edges with label 1 out of thirteen edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most four edges and 1 label for at least nine edges. That is, |ef(0)-ef(1)|=5>1. Hence, for n=4, G is not a prime cordial graph.
For n=5, to satisfy the edge condition for prime cordial labeling it is essential to label seven edges with label 0 and eight edges with label 1 out of fifteen edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most six edges and 1 label for at least nine edges. That is, |ef(0)-ef(1)|=3>1. Hence, for n=5, G is not a prime cordial graph.
Case 2(n=6 to 10). For n=6, f(u1)=4, f(u2)=8 and f(v0)=6, f(v1)=3, f(v2)=9, f(v3)=7, f(v4)=5, f(v5)=1, and f(v6)=2. Then ef(0)=8, ef(1)=9.
For n=7, f(u1)=4, f(u2)=8 and f(v0)=6, f(v1)=3, f(v2)=9, f(v3)=7, f(v4)=5, f(v5)=10, f(v6)=1, and f(v7)=2. Then ef(0)=4, ef(1)=5.
For n=8, f(u1)=11, f(u2)=10 and f(v0)=6, f(v1)=3, f(v2)=2, f(v3)=8, f(v4)=4, f(v5)=5, f(v6)=7, f(v7)=1, and f(v8)=9. Then ef(0)=10, ef(1)=11.
For n=9, f(u1)=11, f(u2)=12 and f(v0)=2, f(v1)=1, f(v2)=3, f(v3)=6, f(v4)=8, f(v5)=4, f(v6)=10, f(v7)=5, f(v8)=7, and f(v9)=9. Then ef(0)=11, ef(1)=12.
For n=10, f(u1)=13, f(u2)=12 and f(v0)=2, f(v1)=9, f(v2)=3, f(v3)=6, f(v4)=4, f(v5)=8, f(v6)=10, f(v7)=5, f(v8)=1, f(v9)=7, and f(v10)=11. Then ef(0)=12, ef(1)=13.
Case 3 (n is even, n≥12). Consider
(14)f(v0)=2,f(v1)=5,f(v2)=10,f(v3)=4,f(v4)=8,f(v4+i)=12+2(i-1),1≤i≤(n/2)-5f(vn/2)=6,f(v(n/2)+1)=3,f(v(n/2)+2)=9,f(vn-1)=1,f(vn)=7,f(v(n/2)+2+i)=11+2(i-1),1≤i≤(n/2)-4f(u1)=n+3,f(u2)=n+2.
In view of the above defined labeling pattern for Case 3, we have ef(0)=n+3 and ef(1)=n+2 for n≡4(mod7) and ef(0)=n+2 and ef(1)=n+3 for n≢4(mod7).
Case 4 (n is odd, n≥11). Consider
(15)f(v0)=2,f(v1)=10,f(v2)=4,f(v3)=8,f(v3+i)=12+2(i-1),1≤i≤((n-1)/2)-4f(v(n-1)/2)=6,f(v(n+1)/2)=3,f(v(n+3)/2)=1,f(vn+1-i)=5+2(i-1),1≤i≤(n-3)/2f(u1)=n+2,f(u2)=n+3.
In view of the above defined labeling pattern for Case 4, we have ef(0)=n+3 and ef(1)=n+2 for n≡3(mod5) and ef(0)=n+2 and ef(1)=n+3 for n≢4(mod7).
In view of Cases 2 to 4 we have |ef(0)-ef(1)|≤1.
Hence, G is a prime cordial graph for n≥6.
Illustration 4. Let G be the graph obtained by duplication of an arbitrary rim edge by an edge in wheel W13. Then |V(G)|=16 and |E(G)|=31. In accordance with Theorem 14, Case 4 will be applicable and the corresponding prime cordial labeling is shown in Figure 3. Here ef(0)=16, ef(1)=15.
Theorem 15.
The graph obtained by duplication of an arbitrary spoke edge by an edge in wheel Wn is prime cordial graph, where n=7 and n≥9.
Proof.
Let v0 be the apex vertex of Wn and let v1,v2,…,vn be the rim vertices. Without loss of generality we duplicate the spoke edge e=v0v1 by an edge e′=u1u2 and call the resultant graph G. Then |V(G)|=n+3 and |E(G)|=3n+2. To define f:V(G)→{1,2,3,…,n+3}, we consider following three cases.
Case 1(n=3 to 6 and n=8). For n=3, to satisfy the edge condition for prime cordial labeling it is essential to label five edges with label 0 and six edges with label 1 out of eleven edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most four edges and 1 label for at least seven edges. That is, |ef(0)-ef(1)|=3>1. Hence, for n=3, G is not a prime cordial graph.
For n=4, to satisfy the edge condition for prime cordial labeling, it is essential to label seven edges with label 0 and seven edges with label 1 out of fourteen edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most four edges and 1 label for at least ten edges. That is, |ef(0)-ef(1)|=6>1. Hence, for n=4, G is not a prime cordial graph.
For n=5, to satisfy the edge condition for prime cordial labeling, it is essential to label eight edges with label 0 and nine edges with label 1 out of seventeen edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most six edges and 1 label for at least eleven edges. That is, |ef(0)-ef(1)|=5>1. Hence, for n=5, G is not a prime cordial graph.
For n=6, to satisfy the edge condition for prime cordial labeling, it is essential to label ten edges with label 0 and ten edges with label 1 out of twenty edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most eight edges and 1 label for at least twelve edges. That is, |ef(0)-ef(1)|=4>1. Hence, for n=6, G is not a prime cordial graph.
For n=8, to satisfy the edge condition for prime cordial labeling, it is essential to label thirteen edges with label 0 and thirteen edges with label 1 out of twenty-six edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most twelve edges and 1 label for at least fourteen edges. That is, |ef(0)-ef(1)|=2>1. Hence, for n=8, G is not a prime cordial graph.
Case 2(n=7, n=9 to 12, and n=14,16,18,20,22). For n=7, f(u1)=2, f(u2)=1 and f(v0)=6, f(v1)=9, f(v2)=3, f(v3)=4, f(v4)=8, f(v5)=10, f(v6)=5, and f(v7)=7. Then ef(0)=12, ef(1)=11.
For n=9, f(u1)=2, f(u2)=1 and f(v0)=6, f(v1)=3, f(v2)=12, f(v3)=4, f(v4)=8, f(v5)=10, f(v6)=5, f(v7)=7, f(v8)=11, and f(v9)=9. Then ef(0)=15, ef(1)=14.
For n=10, f(u1)=2, f(u2)=1and f(v0)=6, f(v1)=3, f(v2)=12, f(v3)=4, f(v4)=8, f(v5)=10, f(v6)=5, f(v7)=7, f(v8)=11, f(v9)=13, and f(v10)=9. Then ef(0)=16, ef(1)=16.
For n=11, f(u1)=2, f(u2)=1and f(v0)=6, f(v1)=5, f(v2)=4, f(v3)=8, f(v4)=10, f(v5)=14, f(v6)=12, f(v7)=9, f(v8)=3, f(v9)=13, f(v10)=11, and f(v11)=7. Then ef(0)=18, ef(1)=17.
For n=12, f(u1)=2, f(u2)=1 and f(v0)=6, f(v1)=5, f(v2)=4, f(v3)=8, f(v4)=10, f(v5)=14, f(v6)=12, f(v7)=9, f(v8)=3, f(v9)=13, f(v10)=15, f(v11)=11, and f(v12)=7. Then ef(0)=19, ef(1)=19.
For n=14, f(u1)=2, f(u2)=1 and f(v0)=6, f(v1)=5, f(v2)=4, f(v3)=8, f(v4)=10, f(v5)=12, f(v6)=14, f(v7)=16, f(v8)=3, f(v9)=9, f(v10)=15, f(v11)=17, f(v12)=13, f(v13)=11, and f(v14)=7. Then ef(0)=22, ef(1)=22.
For n=16, f(u1)=2, f(u2)=1 and f(v0)=6, f(v1)=19, f(v2)=4, f(v3)=8, f(v4)=10, f(v5)=12, f(v6)=14, f(v7)=16, f(v8)=18, f(v9)=3, f(v10)=9, f(v11)=5, f(v12)=7, f(v13)=11, f(v14)=13, f(v15)=15, and f(v16)=17. Then ef(0)=25, ef(1)=25.
For n=18, f(u1)=2, f(u2)=1 and f(v0)=6, f(v1)=21, f(v2)=4, f(v3)=8, f(v4)=10, f(v5)=12, f(v6)=14, f(v7)=16, f(v8)=18,f(v9)=20, f(v10)=3, f(v11)=9, f(v12)=5, f(v13)=7, f(v14)=11, f(v15)=13, f(v16)=15, f(v17)=17, and f(v18)=19. Then ef(0)=28, ef(1)=28.
For n=20, f(u1)=2, f(u2)=1 and f(v0)=6, f(v1)=23, f(v2)=4, f(v3)=8, f(v4)=10, f(v5)=12, f(v6)=14, f(v7)=16, f(v8)=18, f(v9)=20, f(v10)=22, f(v11)=3, f(v12)=9, f(v13)=5, f(v14)=7, f(v15)=11, f(v16)=13, f(v17)=15, f(v18)=17, f(v19)=19, and f(v20)=21. Then ef(0)=31, ef(1)=31.
For n=22, f(u1)=2, f(u2)=1 and f(v0)=6, f(v1)=25, f(v2)=4, f(v3)=8, f(v4)=10, f(v5)=12, f(v6)=14, f(v7)=16, f(v8)=18, f(v9)=20, f(v10)=22, f(v11)=24, f(v12)=3, f(v13)=5, f(v14)=7, f(v15)=9, f(v16)=11, f(v17)=13, f(v18)=15, f(v19)=17, f(v20)=19, f(v21)=21, and f(v22)=23. Then ef(0)=34, ef(1)=34.
For the next case let t1=⌊(n+3)/2⌋, m=⌊(n+3)/3⌋, t2=⌈m/2⌉, k1=⌊(3n+2)/2⌋, k2=2t1+t2-4,
(16)t=k1-k2,t3={t-2;n=13,15,17,t-1;n=19,21,n≥23.
Case 3 (t1-t≥3 (n=13,15,17,19,21 and n≥23)). Consider
(17)f(u1)=2,f(u2)=5,f(v0)=6,f(v1)=3,f(v2)=4,f(vn)=1,f(v2+i)=8+2(i-1),1≤i≤tf(vt+3)={9,ift≡4(mod 7)7,otherwisef(vt+4)={7,ift≡4(mod 7)9,otherwisef(vt+4+i)=9+2i,1≤i≤t3.
For 2t+3<n-1,
(18)f(v2t+3+i)=f(v2t+3)+i,1≤i≤(n-1)-(2t+3).
In view of the above defined labeling pattern for Case 3, we have ef(0)=⌊(3n+2)/2⌋ and ef(1)=⌈(3n+2)/2⌉.
Thus for Cases 2 and 3 we have |ef(0)-ef(1)|≤1.
Hence, G is a prime cordial graph for n=7 and n≥9.
Illustration 5. Let G1 be the graph obtained by duplication of arbitrary spoke edge by an edge of wheel W23. Then |V(G)|=26 and |E(G)|=71. In accordance with Theorem 15 we have t1=13, m=8, t2=4, k1=35, k2=26, and t=9. Here t1-t=4>3 so labeling pattern described in Case 3 will be applicable. The corresponding prime cordial labeling is shown in Figure 4. It is easy to visualise that ef(0)=35, ef(1)=36.
Theorem 16.
DS(Pn) is a prime cordial graph.
Proof.
Consider Pn with V(Pn)={vi:1≤i≤n}. Here V(Pn)=V1∪V2, where V1={vi:2≤i≤n-1} and V2={v1,vn}. Now in order to obtain DS(Pn) from G, we add w1,w2 corresponding to V1,V2. Then |V(DS(Pn))|=n+2 and E(DS(Pn))={v1w2,v2w2}∪{w1vi:2≤i≤n-1}∪E(Pn) so |E(DS(Pn)|=2n-1. We define vertex labeling f:V(DS(Pn))→{1,2,…,n+2} as follows.
Let p1 be the highest prime number <n+2 and k=⌊(n+2)/2⌋. Consider
(19)f(w1)=2,f(w2)=3,f(v1)=1,f(vn)=9,f(vn-1)=p1.
For 0≤i<k-1,
(20)f(v2+i)={(n+2)-2i;niseven(n+1)-2i;nisodd.
And for vertices vk+2,vk+3,…,vn-2 we assign distinct odd numbers (<n+2) in ascending order starting from 5.
In view of the above defined labeling pattern, if n is even number, then ef(0)=n, ef(1)=n-1; otherwise ef(0)=n-1, ef(1)=n.
Thus, |ef(0)-ef(1)|≤1.
Hence, DS(Pn) is a prime cordial graph.
Illustration 6. Prime cordial labeling of the graph DS(P7) is shown in Figure 5.
Theorem 17.
DS(Bn,n) is a prime cordial graph.
Proof.
Consider Bn,n with V(Bn,n)={u,v,ui,vi:1≤i≤n}, where ui,vi are pendant vertices. Here V(Bn,n)=V1∪V2, where V1={ui,vi:1≤i≤n} and V2={u,v}. Now in order to obtain DS(Bn,n) from G, we add w1,w2 corresponding to V1,V2. Then |V(DS(Bn,n)|=2n+4 and E(DS(Bn,n))={uv,uw2,vw2}∪{uui,vvi,w1ui,w1vi:1≤i≤n} so |E(DS(Bn,n)|=4n+3. We define vertex labeling f:V(DS(Bn,n))→{1,2,…,2n+4} as follows:
(21)f(u)=6,f(v)=1,f(w1)=2,f(w2)=3,f(u1)=4,f(ui+1)=8+2(i-1),1≤i≤n-1f(vi)=5+2(i-1),1≤i≤n.
In view of the above defined labeling pattern we have ef(0)=2n+1, ef(1)=2n+2.
Thus, |ef(0)-ef(1)|≤1.
Hence, DS(Bn,n) is a prime cordial graph.
Illustration 7. Prime cordial labeling of the graph DS(B5,5) is shown in Figure 6.
3. Conclusion
A new approach for constructing larger prime cordial graph from the existing prime cordial graph is investigated.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors’ thanks are due to the anonymous referees for careful reading and constructive suggestions for the improvement in the first draft of this paper.
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