International Journal of Computational and Applied . ISSN 1819-4966 Volume 12, Number 1 (2017) © Research India Publications http://www.ripublication.com

Some New Families of Face Cordial Labeling of Graph M. Mohamed Sheriff#1 and A. Farhana Abbas*2 # P.G. and Research Department of Mathematics, Hajee Karutha Rowther Howdia College, Uthamapalayam - 625 533, Tamil Nadu, India. * Research Scholar, School of Mathematics, Madurai Kamaraj University, Madurai - 625 021, Tamil Nadu, India. 1 [email protected] [email protected]

Abstract - In this paper, we have investigated the face integer cordial labeling of gear graph Gn, switching of any one of cycle Cn, PnPm and CnK2.

Keywords - Integer cordial graph, face integer cordial labeling, face integer cordial graph.

I. INTRODUCTION We begin with simple, finite, planar, undirected graph. A (p,q) planar graph G means a graph G=(V,E), where V is the of vertices with |V| = p, E is the set of edges with |E| = q and F is the set of interior faces of G with |F| = number of interior faces of G. For standard terminology and notations related to we refer to Harary [3]. A graph labeling is the assignment of unique identifiers to the edges and vertices of a graph. For a dynamic survey on various graph labeling problems along with an extensive bibliography we refer to Gallian [2]. A mapping 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. 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 vf(i) = number of vertices having label i under f and ef(i) = number of edges having label i under f*. A binary vertex labeling f of a graph G is called a cordial labeling of G if |vf(0) – vf(1)|  1 and |ef(0) – ef(1)|  1. A graph G is cordial if it admits cordial labeling. In [1], Cahit introduced the concept of cordial labeling of graph.   p p ∗ Let G be a simple connected graph with p vertices. Let f:V  2 ,…, 2  or  p ,…, p  as p is even or odd be an injective map, which induces an edge labeling f∗ 2  2  such that f(uv) = 1, if f(u)+f(v)  0 and f(uv) = 0 otherwise. Let ef(i) = number of edges labeled with i, where i = 0 or 1. f is said to be integer cordial if |ef(0) – ef(1)|  1. A graph G is called integer cordial if it admits an integer cordial labeling. Here [–x,…,x] = {t / t is an integer and |t|  x} and [–x,…,x]* = [–x, …, x] – {0}. In [5], Nicholas et al. introduced the concept of integer cordial labeling of graphs and proved that some standard graphs such as cycle Cn, Path Pn, Wheel graph Wn; n > 3, Star graph K1,n, Helm graph Hn, Closed helm graph CHn are integer cordial, Kn is not integer cordial, Kn,n is integer cordial iff n is even and Kn,n\M is integer cordial for any n, where M is a perfect matching of Kn,n.   p p ∗ For a planar graph G, the vertex labeling function is defined as g : V  2 ,…, 2  or  p ,…, p  as p is even or odd be an injective map, which induces an edge labeling 2  2  function g*: E(G){0,1} such that g*(uv) = 1, if g(u) + g(v)  0 and g*(uv) = 0 otherwise ** ** and face labeling function g : F(G)  {0,1} such that g (f) = 1, if g(v1)+…+ g(vn)  0

387 International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 1 (2017) © Research India Publications http://www.ripublication.com

** and g (f) = 0 otherwise, where v1,v2,…,vn are the vertices of face f. g is called face integer cordial labeling of graph G if |eg(0) – eg(1)|  1 and |fg(0) – fg(1)|  1. eg(i) is the * number of edges of G having label i under g and fg(i) is the number of interior faces of G having label i under g** for i = 1,2. A planar graph G is face integer cordial if it admits face integer cordial labeling. In [4], Mohamed Sheriff et al introduced the concept of face integer cordial labeling of graphs and proved the face integer cordial labeling of wheel Wn, fan fn, triangular snake Tn, double triangular snake DTn, star of cycle Cn and DS(Bn,n). The brief summaries of definition which are necessary for the present investigation are provided below. Definition : 1.1

A wheel Wn is a graph with n+1 vertices, formed by connecting a single vertex to all the vertices of cycle Cn. It is denoted by Wn = Cn + K1. Definition : 1.2

The gear graph Gn is obtained from the wheel Wn by subdividing each of its rim edge. Definition : 1.3

The corona G1G2 of two graphs G1(p1, q1) and G2(p2, q2) is defined as the graph th obtained by taking one copy of G1 and p1 copies of G2 and then joining the i vertex of th G1 to all the vertices in the i copy of G2. Definition : 1.4

A vertex switching Gv of a graph G is obtained by taking a vertex v of G, removing the entire edges incident with v and adding edges joining v to every vertex which are not adjacent to v in G.

II. MAIN THEOREMS Theorem : 2.1

The gear graph Gn is a face integer cordial graph for n  3. Proof. Let v be the apex vertex, v1,v2,…,v2n be rim vertices, e1, e2, …, e3n be edges and f1, f2, ..., fn be interior faces of the gear graph Gn, where ei = vv2i–1, for 1  i  n, en+i = vivi+1, for 1  i  2n–1, e3n = v2nv1, fi = vv2i–1v2iv2i+1v, for 1  i  n–1 and fn = vv2n–1v2nv1v. Let G be the gear graph Gn. Then |V(G)| = 2n+1, |E(G)| = 3n and |F(G)| = n. Case (i) : n is odd. Let n = 2k+1. Define vertex labeling of g : V(G)  [–n,…,n] as follows. g(v) = 0 g(vi) = –i for 1  i  n

g(v in ) = n+1– i for 1  i  n Then induced edge labels are * 1n g (ei) = 0 for 1  i  2 *  3n g (ei) = 1 for  i  n 2 * g (en+i) = 0 for 1  i  n–1 * g (en+i) = 1 for n  i  2n

388 International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 1 (2017) © Research India Publications http://www.ripublication.com

Also the induced face labels are ** 1n g (fi) = 0 for 1  i  2 ** 1n g (fi) = 1 for  i  n 2 13n In view of the above defined labeling pattern, we have ef(1) = ef(0)+1 = and 2 1n fg(1) = fg(0) + 1 = . Then |eg(0) − eg(1)|  1 and |fg(0) − fg(1)|  1. 2

Thus the wheel Wn is the face integer cordial for n is odd. Case 2: n is even. Let n = 2k. Define vertex labeling of g : V(G)  [–n,…,n] as follows. g(v) = 0

g(vi) = –i for 1  i  n

g(v in ) = n+1– i for 1  i  n Then induced edge labels are n g*(ei) = 0 for 1  i  2  2n g*(ei) = 1 for  i  n 2 g*(en+i) = 0 for 1  i  n g*(en+i) = 1 for n+1  i  2n Also the induced face labels are n g**(fi) = 0 for 1  i  2  2n g**(fi) = 1 for  i  n 2 3n In view of the above defined labeling pattern, we have ef(0) = ef(1) = and fg(1) = 2 fg(0) = . Then |eg(0) − eg(1)|  1 and |fg(0) − fg(1)|  1.

Thus the gear graph Gn is the face integer cordial for n is even. Hence the gear graph Gn is the face integer cordial graph for n  3. Example : 2.1

The gear graph G5 and its face integer cordial labeling is shown in figure 2.1.

 1 1 1 2 1 0 1 2 1 3

1 1 0 3 1 4 1 4 4 5 5 1 1

389 International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 1 (2017) © Research India Publications http://www.ripublication.com

Figure 2.1 Theorem : 2.2

Switching of any one vertex in cycle Cn admits face integer cordial labeling for n  4. Proof. Let v1,v2,…,vn be the successive vertices of Cn. Gv denotes graph is obtained by switching of vertex v of Cn. Without loss of generality let the switched vertex be v1. Let G be a graph G . Then v1,v2,…,vn are vertices of G, e1,e2,…,e2n–5 are edges of G v1 and f1,f2,…,fn–4 are the interior faces of G. ei = v1vi+2, for 1  i  n–3, en–3+i = vi+1vi+2, for 1  i  n–2. Then |V(G)| = n, |E(G)| = 2n–5 and |F(G)| = n–4. Case (i) : n is even and n = 2k. Define a vertex labeling g : V(G) → [–k,…,k]* as follows n g(vi) = – i, for 1  i  2  2n g(vi) = i – , for  i  n 2 Then induced edge labels are *  4n g (ei) = 0, for 1  i  2 *  2n g (ei) = 1, for  i  n–3 2 *  2n g (en–3+i) = 0, for 1  i  2 * n g (en–3+i) = 1 for  i  n–2 2 Also the induced face labels are ** g (fi) = 0 for 1  i 

**  2n g (fi) = 1 for  i  n–4 2 2n  4 In view of the above defined labeling pattern, we have ef(1) = ef(0)+1 = and 2  4n fg(0) = fg(1) = . 2 Then |eg(0) − eg(1)|  1 and |fg(0) − fg(1)|  1 Hence G is face integer cordial graph for n is even. Case (ii) : n is odd and n = 2k+1. Define a vertex labeling g : V(G) → [–k,…,k] as follows g(v1) = 0, 1n g(vi+1) = – i, for 1  i  2 1n g(vi+1) = i – , for  i  n–1 2 Then induced edge labels are *  3n g (ei) = 0, for 1  i  2

390 International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 1 (2017) © Research India Publications http://www.ripublication.com

* 1n g (ei) = 1, for  i  n–3 2 * 1n g (en–3+i) = 0, for 1  i  2 * 1n g (en–3+i) = 1 for  i  n–2 2 Also the induced face labels are **  3n g (fi) = 0 for 1  i  2 ** 1n g (fi) = 1 for  i  n–4 2 In view of the above defined labeling pattern, we have ef(0) = ef(1)+1 = n–2 and fg(0)  3n = fg(1)+1 = . 2 Then |eg(0) − eg(1)|  1 and |fg(0) − fg(1)|  1 Hence G is face integer cordial graph for n is odd. Therefore switching of any one vertex in cycle Cn is face integer cordial graph for n  4. Example : 2.2

Switching of a vertex v1 in cycle C8 and its face integer cordial labeling is shown in figure 2.2.

1 4 0 2 1

3 3 1 0 3

2 4 1

Figure 2.2

Theorem : 2.3

CnK2 is face integer cordial graph for n  3. Proof. Let v1,v2,...,vn and e1,e2,...,en be the vertices and edges of Cn. Let G be a graph CnK2. Let v1,v2,...,vn,vij for 1  i  n and 1  j  2, e1,e2,...,en,eij for 1  i  n and 1  j  3 and f,f1,f2,...,fn be vertices, edges and an interior faces of G, where ei = vivi+1, for 1  i  n–1, en = vnv1, ei1 = vi vi1, ei2 = vi1vi2, ei3 = vi2vi, for 1  i  n, f = v1v2...vn and fi = vivi1vi2vi for 1  i  n. Then |V(G)| = 3n, |E(G)| = 4n and |F(G)| = n+1. Case (i): n is even. Let 3n = 2k Define a vertex labeling g : V(G)  [–k,…,k]* as follows. n g(vi) = i, for 1  i  2

391 International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 1 (2017) © Research India Publications http://www.ripublication.com

n  2n g(vi) = – i, for  i  n 2 2 n n g(vij) = +2(i–1)+j, for 1  i  and 1  j  2 2 2   2n  g(vij) = – –2 i  –j, for  i  n and 1  j  2  2  Then induced edge labels are * g (ei) = 1, for 1  i 

* g (ei) = 0, for  i  n

* g (eij) = 1, for 1  i  and 1  j  3

* g (eij) = 0, for  i  n and 1  j  3 Also the induced face labels are g**(f) = 1, ** g (fi) = 1 for 1  i 

** g (fi) = 0 for  i  n

In view of the above defined labeling pattern we have eg(1) = eg(0) = 2n and fg(1) =  2n fg(0)+1 = . 2 Then | eg(0) − eg(1)|  1 and | fg(0) − fg(1)|  1 Therefore, the graph CnK2 is face integer product cordial graph for n is even. Case (ii): n is odd. Let 3n = 2k+1 Define a vertex labeling g : V(G)  [–k,…,k] as follows. 1n g(vi) = 1+i, for 1  i  2

g(v 1n ) = 0, 2  1n   3n g(vi) = –1 – i  , for  i  n–1  2  2

g(vn) = – k

g(vij) = 1+ +2(i–1)+j, for 1  i  and 1  j  2 1n g(vi1) = 1, for i = 2 1n g(vi2) = –1, for i = 2   3n   3n g(vij) = –1– –2 i  –j, for  i  n–1 and 1  j  2  2  2

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g(vn1) = –k+1,  1n  g(vn2) = –  .  2  Then induced edge labels are * 1n g (ei) = 1, for 1  i  2 * 1n g (ei) = 0, for  i  n 2 * g (eij) = 1, for 1  i  and 1  j  3

* 1n g (eij) = 1, for i = and 1  j  2 2 * g (ei3) = 0, for i =

*  3n g (eij) = 0, for  i  n and 1  j  3 2 Also the induced face labels are g**(f) = 0, ** 1n g (fi) = 1, for 1  i  2 **  3n g (fi) = 0, for  i  n 2 In view of the above defined labeling pattern we have eg(1) = eg(0) = 2n and fg(1) = 1n fg(0) = . Then |eg(0) − eg(1)|  1 and | fg(0) − fg(1)|  1 2 Therefore, the graph CnK2 is face integer product cordial graph for n is even and m is even. Therefore CnK2 is face integer cordial graph for n  3. Example : 2.3

The graph C5K2 and its face integer cordial labeling of graph is shown in figure 2.3.

1 1

1 4 7

0 5 3 2 6 6

1

7 2 5 6 0 3 4 4 Figure 2.3

393 International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 1 (2017) © Research India Publications http://www.ripublication.com

Theorem : 2.4

The graph PnPm is face integer product cordial graph for any n, m  2. Proof. Let v1,v2,…,vn and e1,e2,…,en–1 be the vertices and edges of Pn. Let G be the graph PnPm. The vertex set V(G) = {vi,vij:1  i  n,1  j  m}, edge set E(G) = {ei,ejk:1  i  n – 1, 1  j  n and 1  k  2m–1} and interior face set F(G) = {fi : 1  i  n(m–1)}, where ei = vivi+1 for 1  i  n–1, ejk = vjvjk for 1  j  n and 1  k  m, ej(m+k) = vjkvj(k+1) for 1  j  n and 1  k  m–1, fij = vi vik vi(k+1) vi for 1  i  n and 1  k  m–1. Then |V(G)| = n(m+1), |E(G)| = 2nm–1 and |F(G)| = n(m–1). Case (i): n is even and m is either odd or even. Let n(m+1) = 2k Define a vertex labeling g : V(G)  [–k,…,k]* as follows. n g(vi) = i, for 1  i  2 n  2n g(vi) = – i, for  i  n 2 2 n g(vij) = +m(i–1)+j, for 1  i  and 1  j  m 2   2n  g(vij) = – –m i  –j, for  i  n and 1  j  m.  2  Then induced edge labels are * g (ei) = 1, for 1  i 

* g (ei) = 0, for  i  n–1

* g (eij) = 1, for 1  i  and 1  j  2m–1

* g (eij) = 0, for  i  n and 1  j  2m–1 Also the induced face labels are ** n(m 1) g (fij) = 1 for 1  i  and 1  j  m 2 ** n(m 1)  2 g (fij) = 0 for  i  n(m–1) and 1  j  m 2 In view of the above defined labeling pattern we have eg(1) = eg(0)+1 = nm and fg(0)

= fg(1) = .

Then |eg(0) − eg(1)|  1 and | fg(0) − fg(1)|  1 Therefore, the graph PnPm is face integer product cordial graph for n is even and m is either odd or even.

Case (ii): n is odd and m is even. Let n(m+1) = 2k+1 Define a vertex labeling g : V(G)  [–k,…,k] as follows.

394 International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 1 (2017) © Research India Publications http://www.ripublication.com

m 1n g(vi) = +i, for 1  i  2 2

g(v 1n ) = 0, 2  1n   3n g(vi) = – – i  , for  i  n  2  2

g(vij) = + +m(i–1)+j, for 1  i  and 1  j  m 1n g(vij) = j, for i = and 1  j  2  m 1n  2m g(vij) = – j  , for i = and  j  m  2  2 2   3n  g(vij) = – – –m i  –j, for  i  n and 1  j  m  2  Then induced edge labels are * g (ei) = 1, for 1  i 

* 1n g (ei) = 0, for  i  n–1 2 * g (eij) = 1, for 1  i  and 1  j  2m–1

* 1n m g (eij) = 1, for i = and 1  j  2 2 *  2m g (eij) = 0, for i = and  j  m 2 * 3m g (eij) = 1, for i = and m+1  j  2 * 3m  2 g (eij) = 0, for i = and  j  2m–1 2 *  3n g (eij) = 0, for  i  n and 1  j  2m–1 2 Also the induced face labels are ** n(m 1)  1 g (fij) = 1 for 1  i  and 1  j  m 2 ** n(m 1)  3 g (fij) = 0 for  i  n(m–1) and 1  j  m 2 In view of the above defined labeling pattern we have eg(1) = eg(0)+1 = nm and fg(1) n(m 1)  1 = fg(0)+1 = . 2 Then |eg(0) − eg(1)|  1 and | fg(0) − fg(1)|  1 Therefore, the graph PnPm is face integer product cordial graph for n is even and m is even.

Case (iii): n is odd and m is odd. Let n(m+1) = 2k

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Define a vertex labeling g : V(G)  [–k,…,k]* as follows. 1m 1n g(vi) = +i, for 1  i  2 2

g(v 1n ) = 1, 2 1m  1n   3n g(vi) = – – i  , for  i  n 2  2  2 1m g(vij) = + +m(i–1)+j, for 1  i  and 1  j  m 2 1n 1m g(vij) = 1+j, for i = and 1  j  2 2   1m  1n 1m g(vij) = – j    , for i = and  j  m   2  2 2   3n  g(vij) = – – –m i  –j, for  i  n and 1  j  m  2  Then induced edge labels are * g (ei) = 1, for 1  i 

* 1n g (ei) = 0, for  i  n–1 2 * g (eij) = 1, for 1  i  and 1  j  2m–1

* 1n 1m g (eij) = 1, for i = and 1  j  2 2 *  3m g (eij) = 0, for i = and  j  m 2 * 3m 1 g (eij) = 1, for i = and m+1  j  2 * 3m 1 g (eij) = 0, for i = and  j  2m–1 2 *  3n g (eij) = 0, for  i  n and 1  j  2m–1 2 Also the induced face labels are ** n(m 1) g (fij) = 1 for 1  i  and 1  j  m 2 ** n(m 1)  2 g (fij) = 0 for  i  n(m–1) and 1  j  m 2 In view of the above defined labeling pattern we have eg(1) = eg(0)+1 = nm and fg(1) n(m 1) = fg(0) = . 2 Then |eg(0) − eg(1)|  1 and | fg(0) − fg(1)|  1 Therefore, the graph PnPm is face integer product cordial graph for n is even and m is even. Hence PnPm is face integer product cordial graph for any n, m  2.

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Example : 2.4

The graph P3P5 and its face product cordial labeling is given in figure 2.4.

5 6 7 8 9 2 3 1 2 3 5 6 7 8 9

1 0

4 1 4 1 0

Figure 2.4

III. CONCLUSION

In this paper, we prove the gear graph Gn, switching of any one vertex of cycle Cn, PnPm and CnK2 are face integer cordial graph.

REFERENCES [1]. I. Cahit, Cordial graphs: A weaker version of graceful and harmonious graphs, Ars Combinatoria, Vol 23, pp. 201-207, 1987. [2]. J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 16, # DS6, 2016. [3]. F. Harary, Graph theory, Addison Wesley, Reading, Massachusetts, 1972. [4]. M. Mohamed Sheriff and A. Farhana Abbas, Face Integer Edge Cordial Labeling of Graphs, communicated. [5]. T.Nicholas and P.Maya, Some results on integer cordial graph, Journal of Progressive Research in Mathematics (JPRM), Vol. 8, Issue 1, pp 1183-1194, 2016.

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