412 P. Jeyanthi, S. Philo and Maged Z. Youssef

1. Introduction

Throughout this paper by a graph we mean a finite, simple and undirectedone. For standard terminology and notation we follow Harary [3]. A graphG(V,E) with p vertices and q edges is called a (p, q) — graph. The graphlabeling is an assignment of integers to the set of vertices or edges or both,subject to certain conditions. An extensive survey of various graph labelingproblems is available in [1]. Graham and Sloane [2] introduced harmoniouslabeling during their study of modular versions of additive bases problemsstemming from error correcting codes. A graph G is said to be harmo-nious if there exists an injection f : V (G) → Zq such that the inducedfunction f∗ : E(G)→ Zq defined by f

∗(uv) = (f(u) + f(v)) (mod q) is abijection and f is called harmonious labeling of G. The concept of an oddharmonious labeling was due to Liang and Bai [4]. A graph G is said to beodd harmonious if there exists an injection f : V (G)→ {0, 1, 2, · · · , 2q − 1}such that the induced function f∗ : E(G) → {1, 3, · · · , 2q − 1} defined byf∗(uv) = f(u) + f(v) is a bijection. If f(V (G)) = {0, 1, 2, ....q} then f iscalled as strongly odd harmonious labeling and G is called as strongly oddharmonious graph.The odd harmoniousness of graph is useful for the solu-tion of undetermined equations. The following results have been publishedin [4].

1. If G is an odd harmonious graph, then G is a bipartite graph. Henceany graph that contains an odd cycle is not odd harmonious.

2. If a (p, q)-graph G is odd harmonious, then 2√q ≤ p ≤ (2q − 1).

3. If G is an odd harmonious Eulerian graph with q edges, then q ≡0, 2(mod 4).

Jeyanthi et al. [5]-[12] proved that the shadow and splitting of graphK2,n , Cn for n ≡ 0(mod 4), Hn,n, any two even cycles sharing a com-mon vertex and a common edge, plus graph Pln , open star of plus graphS(t.P ln), path union of plus graph Pln, joining of Cm and plus graph Plnwith a path, one point union of path of plus graph P t

n(t.n.P lm), super subdi-vision of any cycle Cm with m ≥ 3 ,ladder, cycle Cn for n ≡ 0(mod 4) withK1,m, m-shadow and m-splitting of the graphs Pn, Hn,n, Kr,s, Pn ⊕ K2

and Splm(Cn), n ≡ 0(mod 4) subdivided shell graphs, Sm,n, spiders, m-

Odd harmonious labeling of grid graphs 413

shadow and m-splitting graphs are odd harmonious graphs. Selvaraju etal. [13] proved that K1,n, S(K1,n), Tm,n,t and S(Tm,n,t) are odd harmo-nious. Vaidya and Shah [14], [15] proved that shadow and splitting graphsof Pn, K1,n,Bn,n and super subdivision of Hn,n are odd harmonious.

We use the following definitions in the subsequent section.

Definition 1. The Cartesian product of graphsG andH denoted asG×H,is the graph with vertex set V (G) × V (H) = {(u, v)/u ∈ V (G) and v ∈V (H)} and (u, v) is adjacent to (u,, v,) if and only if either u = u, and edge(v, v,) ∈ E(H) or v = v, and edge (u, u,) ∈ E(G). The Cartesian productof two paths Pm and Pn denoted by Pm × Pn is known as a grid graph onmn vertices and 2mn− (m+ n) edges.

Definition 2. Let G be a graph and G1, G2, · · · , Gn, n ≥ 2 be n copiesof graph G.Then the graph obtained by adding an edge from Gi to Gi+1

(i = 1, 2, · · · , n− 1) is called path union of graph G.

Definition 3. A graph G is obtained by replacing each edge of K1,t by apath Pn of length n on n+ 1 vertices is called one point union for t copiesof path Pn, denoted by P

tn.

Definition 4. A graph G is obtained by replacing each vertices of P tn ex-

cept the central vertex by the graphs G1, G2, · · · , Gn is known as one pointunion for path of graphs, denoted by P t

n(G1, G2, · · · , Gn) where Ptn is the one

point union of t copies of path Pn. If we replace each vertices of Ptn except

the central vertex by the graph H, that is G1 = G2 = G3 = · · ·Gn = H,such one point union of path graph, denoted by P t

n(t.n.H).

Definition 5. Let G = (V,E) be a graph with p vertices and q edges. Agraph H is said to be a t-super subdivision of G if H is obtained from Gby replacing every edge e of G by a complete bipartite graph K2,t for somet ∈ N .

2. Main Results

Theorem 2.1. Path union of t copies of Pm×Pn is odd harmonious, wherem,n, t ≥ 2.

414 P. Jeyanthi, S. Philo and Maged Z. Youssef

Proof. Let G be a path union of t copies of Pm × Pn, ∀m,n, t ≥ 2.

Let xk,i,1, xk,i,2, · · · , xk,i,n be the vertices of the kth copy of ith row ver-tices in G, where 1 ≤ k ≤ t, 1 ≤ i ≤ m. Now, we join these consecutivecopies of the grid graph Pm × Pn by an edge. Join xk,m,n with xk+1,1,1,∀1 ≤ k ≤ t−1 by an edge to form a path union of t copies of the grid grpahPm × Pn.

Then |V (G)| = tmn and |E(G)| = t[2mn− (m+ n) + 1]− 1.

We define a labeling f : V (G)→ {0, 1, 2, · · · , 2{t[2mn− (m+ n) + 1]− 1}− 1}as follows:

f(xk,i,j) = (i−1)(2n−1)+j−1+(k−1)[m(2n−1)−n+1], 1 ≤ i ≤ m,1 ≤ j ≤ n, 1 ≤ k ≤ t.The induced edge labels are

f∗(xk,i,jxk,i,j+1) = 2(i−1)(2n−1)+2j−1+2(k−1)[m(2n−1)−n+1],1 ≤ i ≤ m, 1 ≤ j ≤ n− 1, 1 ≤ k ≤ t;

f∗(xk,i,jxk,i+1,j) = 2i(2n−1)+2(j−1)−2n+1+2(k−1)[m(2n−1)−n+1],1 ≤ i ≤ m− 1, 1 ≤ j ≤ n, 1 ≤ k ≤ t;

f∗(xk,m,nxk+1,1,1) = 2k[m(2n− 1)− n+ 1]− 1, 1 ≤ k ≤ t− 1.

In view of the above defined labeling pattern, path union of t copies ofPm × Pn is odd harmonious, where m,n, t ≥ 2 .2

An odd harmonious labeling of path union of 3 copies of P3 × P4 isshown in Figure 1.

Odd harmonious labeling of grid graphs 415

Theorem 2.2. Path union of t different copies of Pmi × Pni , 1 ≤ i ≤ t isodd harmonious.

Proof. Let G be a path union of t different copies of the grid graphPmi × Pni , 1 ≤ i ≤ t. Let xk,i,1, xk,i,2, · · · , xk,i,nk be the vertices of the kthcopy of the ith row vertices in G, where 1 ≤ i ≤ mk, 1 ≤ k ≤ t. Now, wejoin these consecutive copies of the grid graph Pmi × Pni , 1 ≤ i ≤ t by anedge. Join xk,mk,nk with xk+1,1,1, ∀1 ≤ k ≤ t− 1 by an edge to form a pathunion of t copies of the grid graph Pmi × Pni , 1 ≤ i ≤ t.

Then |V (G)| = n1m1 + n2m2 + · · · + ntmt and |E(G)| = 2(m1n1 +m2n2+ · · ·+mtnt)− (m1+m2+ · · ·+mt)− (n1+n2+ · · ·+nt) + (t− 1).

We define a labeling f : V (G)→ {0, 1, 2, · · · , 2|E(G)|− 1} as follows:

f(xk,i,j) = 2[m1n1+m2n2+ · · ·+mk−1nk−1]− [m1+m2+ · · ·+mk−1]−[n1+n2+ · · ·+ nk−1] + 2nk(i− 1) + j − i+ (k− 1), 1 ≤ k ≤ t, 1 ≤ i ≤ mk,1 ≤ j ≤ nk.

The induced edge labels are

f∗(xk,i,jxk,i,j+1) = 4[m1n1+m2n2+ · · ·+mk−1nk−1]−2[m1+m2+ · · ·+mk−1]− 2[n1 + n2 + · · ·+ nk−1] + 4nk(i− 1) + 2j − 2i+ 2k − 1, 1 ≤ k ≤ t,1 ≤ i ≤ mk, 1 ≤ j ≤ nk − 1;

416 P. Jeyanthi, S. Philo and Maged Z. Youssef

f∗(xk,i,jxk,i+1,j) = 4[m1n1+m2n2+ · · ·+mk−1nk−1]−2[m1+m2+ · · ·+mk−1]− 2[n1+n2+ · · ·+ nk−1] + 2nk(2i− 1)+ 2j − 2i+2k− 3, 1 ≤ k ≤ t,1 ≤ i ≤ mk − 1, 1 ≤ j ≤ nk;

f∗(xk,mk,nkxk+1,1,1) = 4[m1n1+m2n2+ · · ·+mk−1nk−1]−2[m1+m2+· · ·+mk−1]−2[n1+n2+· · ·+nk−1]+4mknk−2mk−2nk+2k−1, 1 ≤ k ≤ t−1.

In view of the above defined labeling pattern, path union of t differentcopies of Pmi × Pni , 1 ≤ i ≤ t is odd harmonious .2

An odd harmonious labeling of path union of 2 copies P5×P4 and P6×P8is shown in Figure 2.

Theorem 2.3. The vertex union of t copies of Pm×Pn is odd harmonious,where m,n ≥ 2.

Proof. Let G be a vertex union of t copies of the grid graph Pm × Pn,∀m,n, t ≥ 2. Let xk,i,1, xk,i,2, · · · , xk,i,n be the vertices of the kth copy ofthe ith row vertices in G, where 1 ≤ k ≤ t, 1 ≤ i ≤ m. Now, we join theseconsecutive copies of the grid graph Pm × Pn by a vertex. Identifying thevertices xk,m,n with xk+1,1,1, ∀1 ≤ k ≤ t − 1 to form a vertex union of tcopies of the grid graph Pm × Pn.

Odd harmonious labeling of grid graphs 417

Case (i): When m = nThen |V (G)| = tn2 − (t− 1) and |E(G)| = 2n(n− 1)t.We define a labeling f : V (G)→ {0, 1, 2, · · · , 2[2n(n− 1)t]− 1} as follows:

f(xk,i,j) = (i − 1)(2n − 1) + j − 1 + (k − 1)[2n(n − 1)], 1 ≤ i, j ≤ n,1 ≤ k ≤ t.

The induced edge labels are

f∗(xk,i,jxk,i,j+1) = 2(i−1)(2n−1)+2j−1+2(k−1)[2n(n−1)], 1 ≤ i ≤ n,1 ≤ j ≤ n− 1, 1 ≤ k ≤ t;

f∗(xk,i,jxk,i+1,j) = 2i(2n− 1) + 2(j − 1)− 2n+1+ 2(k− 1)[2n(n− 1)],1 ≤ i ≤ n− 1, 1 ≤ j ≤ n, 1 ≤ k ≤ t.

Case(ii): When m 6= nThen |V (G)| = tmn− (t− 1) and |E(G)| = [(m− 1)(2n− 1) + n− 1]t.

We define a labeling f : V (G)→ {0, 1, 2, · · · , 2[(m− 1)(2n− 1) + n− 1]t− 1}as follows:

f(xk,i,j) = (i − 1)(2n − 1) + j − 1 + (k − 1)[(m − 1)(2n − 1) + n − 1],1 ≤ i ≤ m, 1 ≤ j ≤ n, 1 ≤ k ≤ t.The induced edge labels are

f∗(xk,i,jxk,i,j+1) = 2(i−1)(2n−1)+2j−1+2(k−1)[(m−1)(2n−1)+n−1],1 ≤ i ≤ m, 1 ≤ j ≤ n− 1, 1 ≤ k ≤ t;

f∗(xk,i,jxk,i+1,j) = 2i(2n−1)+2(j−1)−2n+1+2(k−1)[(m−1)(2n−1) + n− 1], 1 ≤ i ≤ m− 1, 1 ≤ j ≤ n, 1 ≤ k ≤ t.

In view of above defined labeling pattern,the vertex union of t copies ofPm × Pn is odd harmonious, where m,n ≥ 2.

2

An odd harmonious labeling of vertex union of 3 copies of P3 × P3(m = n), is shown in Figure 3.

418 P. Jeyanthi, S. Philo and Maged Z. Youssef

An odd harmonious labeling of vertex union of 2 copies of P3 × P4(m 6= n) is shown in Figure 4.

Odd harmonious labeling of grid graphs 419

Theorem 2.4. The vertex union of t different copies of the grid graphPmi × Pni , 1 ≤ i ≤ t is odd harmonious.

Proof. Let G be a vertex union of t different copies of the grid graphPmi × Pni , 1 ≤ i ≤ t. Let xk,i,1, xk,i,2, · · · , xk,i,nk be the vertices of the kthcopy of the ith row vertices in G, where 1 ≤ i ≤ mk, 1 ≤ k ≤ t. Now, wejoin these consecutive copies of the grid graph Pmi × Pni , 1 ≤ i ≤ t by avertex. Identifying the vertices xk,mk,nk with xk+1,1,1 , ∀1 ≤ k ≤ t − 1 toform a vertex union of t copies of the grid graph Pmi × Pni , 1 ≤ i ≤ t.Then |V (G)| = (n1m1 + n2m2 + · · ·+ ntmt)− (t− 1) and

|E(G)| = 2(m1n1+m2n2+ · · ·+mtnt)− (m1+m2+ · · ·+mt)− (n1+n2 + · · ·+ nt).

We define a labeling f : V (G)→ {0, 1, 2, · · · , 2|E(G)|− 1} as follows:

f(xk,1,j) = 2[m1n1+m2n2+ · · ·+mk−1nk−1]− [m1+ · · ·+mk−1]− [n1+· · ·+ nk−1] + (j − 1), 1 ≤ j ≤ nk, 1 ≤ k ≤ t;

f(xk,i,j) = (2nk−1)(i−1)+ j−1+2[m1n1+m2n2+ · · ·+mk−1nk−1]−[m1 + · · ·+mk−1]− [n1 + · · ·+ nk−1], 1 ≤ j ≤ nk, 2 ≤ i ≤ mk, 1 ≤ k ≤ t.

The induced edge labels are

f∗(xk,1,jxk,1,j+1) = 4[m1n1 + · · · +mk−1nk−1] − 2[m1 + · · · +mk−1] −2[n1 + · · ·+ nk−1] + 2j − 1, 1 ≤ j ≤ nk, 1 ≤ k ≤ t;

f∗(xk,1,jxk,2,j) = (2nk − 1) + 2(j − 1) + 4[m1n1 + · · · + mk−1nk−1] −2[m1 + · · ·+mk−1]− 2[n1 + · · ·+ nk−1], 1 ≤ j ≤ nk, 1 ≤ k ≤ t;

f∗(xk,i,jxk,i,j+1) = 2(2nk− 1)(i− 1)+2j− 1+4[m1n1+ · · ·+mk−1nk−1]−2[m1+· · ·+mk−1]−2[n1+· · ·+nk−1], 2 ≤ i ≤ mk, 1 ≤ j ≤ nk−1, 1 ≤ k ≤ t;

f∗(xk,i,jxk,i+1,j) = (2nk−1)(2i−1)+2(j−1)+4[m1n1+ · · ·+mk−1nk−1]−

420 P. Jeyanthi, S. Philo and Maged Z. Youssef

2[m1+· · ·+mk−1]−2[n1+· · ·+nk−1], 2 ≤ i ≤ mk−1, 1 ≤ j ≤ nk, 1 ≤ k ≤ t.

In the view of above defined labeling pattern, vertex union of t differentcopies of the grid graph Pmi × Pni , 1 ≤ i ≤ t is odd harmonious. 2

An odd harmonious labeling of vertex union of 3 different copies ofP3 × P3, P3 × P4, P4 × P5 is shown in Figure 5.

Theorem 2.5. The t-super subdivision of grid graph Pm×Pn is odd har-monious, where m,n ≥ 2.

Proof. Let xi,j (1 ≤ i ≤ n, 1 ≤ j ≤ m) be the vertices of the grid graphPm × Pn. We know that the number of vertices in Pm × Pn is p = mn andthe number of edges q = 2mn − (m + n). Let G be a graph obtained byt-super subdivision of Pm×Pn. Then we see that the number of vertices inG is P = |V (G)| = p+tq and the number of edges in G is Q = |E(G)| = 2tq.

Let ui,j,k(1 ≤ i ≤ n − 1, 1 ≤ j ≤ m, 1 ≤ k ≤ t) be vertices for thevertical edges in G and vi,j,k(1 ≤ i ≤ n, 1 ≤ j ≤ m − 1, 1 ≤ k ≤ t) be

Odd harmonious labeling of grid graphs 421

vertices for horizontal edges in G.

We define the labeling f : V (G)→ {0, 1, 2, · · · , 4tq − 1} as follows:

f(xi,j) = 2t(i− 1)(2m− 1) + 2t(j − 1), if i is odd, 1 ≤ j ≤ m;

f(xi,j) = 2t(i− 1)(2m− 1) + 2t(m− j), if i is even, 1 ≤ j ≤ m;

f(v1,j,k) = 2t(j − 1) + 2k − 1, 1 ≤ j ≤ m− 1, 1 ≤ k ≤ t;

f(vi,j,k) = 2t(i − 2)(2m − 1) + 2t(m − 1) + 2t − 1 + 2k + 4t(j − 1),1 ≤ j ≤ m− 1, i = 3, 5, · · ·, 1 ≤ k ≤ t;

f(vi,j,k) = 2t(i− 2)(2m− 1) + 2t(m− 1) + 2t− 1 + 2k + 4t(m− j − 1),1 ≤ j ≤ m− 1, i = 2, 4, · · ·, 1 ≤ k ≤ t;

f(ui,j,k) = 2t(i−1)(2m−1)+2t(m−1)+2k−1+4t(m−j), 1 ≤ j ≤ m,i = 3, 5, · · ·, 1 ≤ k ≤ t;

f(ui,j,k) = 2t(i−1)(2m−1)+2t(m−1)+2k−1+4t(j−1), 1 ≤ j ≤ m,i = 2, 4, · · ·, 1 ≤ k ≤ t.

The induced edge labels are

f∗(x1,jv1,s,k) = 2t(j−1)+2t(s−1)+2k−1, 1 ≤ j ≤ m, 1 ≤ s ≤ m−1;

f∗(x1,ju1,s,k) = 2t(j− 1)+2t(m− 1)+2k−1+4t(m− s), 1 ≤ j, s ≤ m;

f∗(xi,jul,s,k) = 2t(i−1)(2m−1)+2t(m−j)+2t(l−1)(2m−1)+2t(m−1) + 2k− 1+ 4t(m− s), i = 2, 4, · · · , 1 ≤ j, s ≤ m, l = 1, 3, 5, · · ·,1 ≤ k ≤ t;

f∗(xi,jul,s,k) = 2t(i−1)(2m−1)+2t(m−j)+2t(l−1)(2m−1)+2t(m−1) + 2k − 1 + 4t(s− 1), i = 2, 4, · · · , 1 ≤ j, s ≤ m, l = 2, 4, 6, · · ·,1 ≤ k ≤ t;

f∗(xi,jul,s,k) = 2t(i−1)(2m−1)+2t(j−1)+2t(l−1)(2m−1)+2t(m−1) + 2k− 1+ 4t(m− s), i = 1, 3, · · · , 1 ≤ j, s ≤ m, l = 1, 3, 5, · · ·,1 ≤ k ≤ t;

f∗(xi,jul,s,k) = 2t(i−1)(2m−1)+2t(j−1)+2t(l−1)(2m−1)+2t(m−1) + 2k − 1 + 4t(s− 1), i = 1, 3, · · · , 1 ≤ j, s ≤ m, l = 2, 4, · · ·,1 ≤ k ≤ t;

422 P. Jeyanthi, S. Philo and Maged Z. Youssef

f∗(xi,jvl,s,k) = 2t(i−1)(2m−1)+2t(j−1)+2t(l−2)(2m−1)+2t(m−1) + 2t − 1 + 2k + 4t(s − 1), i = 1, 3, · · · , 1 ≤ j ≤ m, 1 ≤ s ≤ m − 1,l = 3, 5, · · ·,1 ≤ k ≤ t;

f∗(xi,jvl,s,k) = 2t(i−1)(2m−1)+2t(j−1)+2t(l−2)(2m−1)+2t(m−1) + 2t− 1 + 2k + 4t(m− s− 1), i = 3, 5, · · · , 1 ≤ j ≤ m, 1 ≤ s ≤ m− 1,l = 2, 4, · · ·,1 ≤ k ≤ t;

f∗(xi,jvl,s,k) = 2t(i−1)(2m−1)+2t(m−j)+2t(l−2)(2m−1)+2t(m−1) + 2t − 1 + 2k + 4t(s − 1), i = 2, 4, · · · , 1 ≤ j ≤ m, 1 ≤ s ≤ m − 1,l = 3, 5, · · ·,1 ≤ k ≤ t;

f∗(xi,jvl,s,k) = 2t(i−1)(2m−1)+2t(m−j)+2t(l−2)(2m−1)+2t(m−1) + 2t− 1 + 2k + 4t(m− s− 1), i = 2, 4, · · · , 1 ≤ j ≤ m, 1 ≤ s ≤ m− 1,l = 2, 4, · · ·,1 ≤ k ≤ t.

In the view of above defined labeling pattern, t-super subdivision ofPm × Pn, m,n ≥ 2 is odd harmonious.2

An odd harmonious labeling of 2-super subdivision of P3 × P3 is shownin Figure 6.

Odd harmonious labeling of grid graphs 423

Theorem 2.6. One point union of path of graph P tn(t.n.Pm×Pm), n ≥ 1,

m ≥ 2 is odd harmonious if t is odd.

Proof. Let G = P tn(t.n.Pm × Pm) be a graph obtained by replacing

each vertices of P tn except the central vertex by the graph Pm × Pm. That

means G is the graph obtained by replacing each vertices of K1,t exceptthe apex vertex by the path union of n copies of the graph Pm × Pm.Let u0 be the central vertex for the graph G with t branches. Let xs,k,i,j(∀1 ≤ k ≤ n, 1 ≤ i, j ≤ m) be the vertices of kth copy of path union of ncopies of Pm × Pm lies in the sth branch of the graph G, ∀s = 1, 2, · · · , t.

424 P. Jeyanthi, S. Philo and Maged Z. Youssef

Join the vertices of us,1,1,1 with u0 by an edge to form the one pointunion for path of grid graph G. Also join the vertices us,k,m,m to us,k+1,1,1for k = 1, 2, · · · , n − 1, s = 1, 2, · · · , t by an edge. This graph G with|V (G)| = tnm2 + 1 and |E(G)| = tn[2m(m− 1) + 1].

We define the labeling f : V (G)→ {0, 1, 2, · · · , 2tn[2m(m− 1) + 1]− 1}as follows:f(u0) = 0;

f(xs,k,i,j) = 2 + 4(t − s) + t(j − 1) + t(2m− 1)(i− 1) + 2tm(m− 1) +t(k − 2)[2m(m− 1) + 1], if both i and j are odd, k = 2, 4, · · ·, 1 ≤ s ≤ t;

f(xs,k,i,j) = 2s − 1 + t(j − 2) + t(2m− 1)(i − 1) + t[2m(m− 1) + 2] +t(k − 2)[2m(m− 1) + 1], if i is odd and j is even, k = 2, 4, · · ·, 1 ≤ s ≤ t;

f(xs,k,i,j) = 2mt+1+ t(j − 1)+ t(2m− 1)(i− 2) + 2(s− 1)+ 2tm(m−1)+t(k−2)[2m(m−1)+1], if i is even and j is odd, k = 2, 4, · · ·, 1 ≤ s ≤ t;

f(xs,k,i,j) = 2[1+t(m−1)]+t(j−2)+4(t−s)+t(2m−1)(i−2)+t[2m(m−1) + 2] + t(k − 2)[2m(m − 1) + 1], if both i and j are even, k = 2, 4, · · ·,1 ≤ s ≤ t;

f(xs,k,i,j) = 2s−1+ t(j−1)+ t(2m−1)(i−1)+ t(k−1)[2m(m−1)+1],if both i and j are odd, k = 1, 3, · · ·, 1 ≤ s ≤ t;

f(xs,k,i,j) = 2+4(t−s)+t(j−2)+t(2m−1)(i−1)+t(k−1)[2m(m−1)+1],if i is odd and j is even, k = 1, 3, · · ·, 1 ≤ s ≤ t;

f(xs,k,i,j) = 2[1 + t(m − 1)] + t(j − 1) + 4(t − s) + t(2m − 1)(i − 2) +t(k − 1)[2m(m− 1) + 1], if i is even and j is odd, k = 1, 3, · · ·, 1 ≤ s ≤ t;

f(xs,k,i,j) = 2mt + 1 + t(j − 2) + t(2m − 1)(i − 2) + 2(s − 1) + t(k −1)[2m(m− 1) + 1], if both i and j are even, k = 1, 3, · · ·, 1 ≤ s ≤ t.

The induced edge labels are

f∗(xs,k,i,jxs,k,i,j+1) = 1−2s+4t+2t(j−1)+2t(2m−1)(i−1)+2t(k−1)[2m(m− 1) + 1], if k and i are odd, 1 ≤ j ≤ m− 1, 1 ≤ s ≤ t;

Odd harmonious labeling of grid graphs 425

f∗(xs,k,i,jxs,k,i,j+1) = 1− 2s+ 2t(2m+ 1) + 2t(j − 1) + 2t(2m− 1)(i−2) + 2t(k − 1)[2m(m − 1) + 1], if k is odd and i is even, 1 ≤ j ≤ m − 1,1 ≤ s ≤ t;

f∗(xs,k,i,jxs,k,i,j+1) = 1−2s+6t+2t(j−1)+2t(2m−1)(i−1)+4tm(m−1) + 2t(k − 2)[2m(m − 1) + 1], if k is even and i is odd, 1 ≤ j ≤ m − 1,1 ≤ s ≤ t;

f∗(xs,k,i,jxs,k,i,j+1) = 1 − 2s + 4t + 4tm2 + 2t(j − 1) + 2t(2m − 1)(i −2) + 2t(k − 2)[2m(m − 1) + 1], if both k and i are even, 1 ≤ j ≤ m − 1,1 ≤ s ≤ t;

f∗(xs,k,i,jxs,k,i+1,j) = 1 − 2s + 2t(m + 1) + 2t(j − 1) + 2t(2m − 1)(i −1)+ 2t(k− 1)[2m(m− 1)+ 1], if k and j are odd, 1 ≤ i ≤ m− 1, 1 ≤ s ≤ t;

f∗(xs,k,i,jxs,k,i+1,j) = 1 − 2s + 2t(m + 2) + 2t(j − 2) + 2t(2m − 1)(i −1) + 2t(k − 1)[2m(m − 1) + 1], if k is odd and j is even, 1 ≤ i ≤ m − 1,1 ≤ s ≤ t;

f∗(xs,k,i,jxs,k,i+1,j) = 1−2s+2t(m+2)+2t(j−1)+2t(2m−1)(i−1)+4tm(m−1)+2t(k−2)[2m(m−1)+1], if kis even and j is odd, 1 ≤ i ≤ m−1,1 ≤ s ≤ t;f∗(xs,k,i,jxs,k,i+1,j) = 1− 2s+6t+2tm+2t(j− 2)+4tm(m− 1)+2t(2m−1)(i − 1) + 2t(k − 2)[2m(m − 1) + 1], if k and j are even, 1 ≤ i ≤ m − 1,1 ≤ s ≤ t;f∗(xs,k,m,mxs,k+1,1,1) = 1−2s+4t+4tm(m−1)+2t(k−1)[2m(m−1)+1],if m is odd, 1 ≤ k ≤ m− 1, 1 ≤ s ≤ t;

f∗(xs,k,m,mxs,k+1,1,1) = 1− 2s+2t+ tm(2m+1)+ t(m− 2)(2m− 1) +2t(k − 1)[2m(m− 1) + 1], if m is even, 1 ≤ k ≤ m− 1, 1 ≤ s ≤ t.

In the view of above defined labeling pattern, one point union of pathof grid graph P t

n(t.n.Pm×Pm), n ≥ 1, m ≥ 2 is odd harmonious if t is odd.2

An odd harmonious labeling of one point union of path of grid graphP 32 (3.2.P4 × P4) is shown in Figure 7.

426 P. Jeyanthi, S. Philo and Maged Z. Youssef