TOTAL MEAN CORDIAL LABELING OF SOME CYCLE RELATED GRAPHS

Abstract

A Total Mean Cordial labeling of a graph G = (V, E) is a function $f:V(G){\rightarrow}\{0,1,2\}$ such that $f(xy)={\Large\lceil}\frac{f(x)+f(y)}{2}{\Large\rceil}$ where $x,y{\in}V(G)$, $xy{\in}E(G)$, and the total number of 0, 1 and 2 are balanced. That is ${\mid}ev_f(i)-ev_f(j){\mid}{\leq}1$, $i,j{\in}\{0,1,2\}$ where $ev_f(x)$ denotes the total number of vertices and edges labeled with x (x = 0, 1, 2). If there is a total mean cordial labeling on a graph G, then we will call G is Total Mean Cordial. Here, We investigate the Total Mean Cordial labeling behaviour of prism, gear, helms.

1. Introduction

Terminology and notations in graph theory we refer Harary [2]. New terms and notations shall, however, be specifically defined whenever necessary. By a graph G = (V,E) we mean a finite, undirected graph with neither loops nor mul-tiple edges. The product graph G1×G2 is defined as follows: Consider any two points u = (u1, u2) and v = (v1, v2) in V = V1×V2. Then u and v are adjacent in G1×G2 whenever [u1 = v1 and u2 adj v2] or [u2 = v2 and u1 adj v1]. The join of two graphs G1 and G2 is denoted by G1 + G2 and whose vertex set is V (G1 + G2) = V (G1) ∪ V (G2) and edge set E (G1 + G2) = E (G1) ∪ E (G2) ∪ {uv : u ∈ V (G1), v ∈ V (G2)}. The order and size of G are denoted by p and q respectively. Ponraj, Ramasamy and Sathish Narayanan [3] introduced the concept of Total Mean Cordial labeling of graphs and studied about their be-havior on Path, Cycle, Wheel and some more standard graphs. In [4], Ponraj and Sathish Narayanan proved that is Total Mean Cordial if and only if n = 1 or 2 or 4 or 6 or 8. Also in [5], Ponraj, Ramasamy and Sathish Narayanan studied about the Total Mean Cordiality of Lotus inside a circle, bistar, flower graph, K2,n, Olive tree, In this paper, we investigate the Total Mean Cordiality of some cycle related graphs. Let x be any real number. Then the symbol ⌈x⌉ stands for the smallest integer greater than or equal to x.

 

2. Main results

Definition 2.1. A Total Mean Cordial labeling of a graph G = (V,E) is a function f : V (G) → {0, 1, 2} such that where x, y ∈ V (G), xy ∈ E (G), and the total number of 0, 1 and 2 are balanced. That is |evf(i) - evf(j)| ≤ 1, i, j ∈ {0, 1, 2} where evf (x) denotes the total number of vertices and edges labeled with x (x = 0, 1, 2). If there exists a total mean cordial labeling on a graph G, we will call G is Total Mean Cordial.

Prisms are graphs of the form Cm × Pn. We now look into the graph prism Cn × P2.

Theorem 2.2. Prisms are Total Mean Cordial.

Proof. It is clear that p + q = 5n. Let V (Cn × P2) = {ui, vi : 1 ≤ i ≤ n} and E(Cn×P2) = {u1un, v1vn}∪{uivi : 1 ≤ i ≤ n}∪{uiui+1, vivi+1 : 1 ≤ i ≤ n−1}.

Case 1. n ≡ 0 (mod 6).

Let n = 6t and t > 0. Define a map f : V (Cn × P2) → {0, 1, 2} by

f(u2t+1) = 0, f(v2t+1) = f(v6t) = 2, f(u6t) = 1. In this case evf (0) = evf (1) = evf (2) = 10t.

Case 2. n ≡ 1 (mod 6).

Let n = 6t + 1 and t > 0. Define a map f : V (Cn × P2) → {0, 1, 2} by

f(v2t+1) = f(v2t+2) = 1. Here evf (0) = evf (2) = 10t + 2, evf (2) = 10t + 1.

Case 3. n ≡ 2 (mod 6).

Let n = 6t + 2 and t > 0. Define a map f : V (Cn × P2) → {0, 1, 2} by

f(u6t+2) = 1, f(v6t+2) = 2. In this case evf (0) = evf (2) = 10t + 3, evf (1) = 10t + 4.

Case 4. n ≡ 3 (mod 6).

Let n = 6t + 3 and t ≥ 0. A Total Mean Cordial labeling of C3 × P2 is given in figure 1.

FIGURE 1.

Assume t > 0. Define a map f : V (Cn × P2) → {0, 1, 2} by

f(u2t+2) = 0, f(u6t+3) = 1, f(v2t+2) = f(v6t+3) = 2. In this case evf (0) = evf (1) = evf (2) = 10t + 5.

Case 5. n ≡ 4 (mod 6).

Let n = 6t + 4 and t ≥ 0. A Total Mean Cordial labeling of C4 × P2 is given in figure 2.

FIGURE 2.

Assume t > 0. Define a map f : V (Cn × P2) → {0, 1, 2} by

In this case evf (0) = evf (1) = 10t + 7, evf (2) = 10t + 6.

Case 6. n ≡ 5 (mod 6).

Let n = 6t - 1 and t > 0. Define a map f : V (Cn × P2) → {0, 1, 2} by

f(u6t-1) = 1, f(v6t-1) = 2. In this case evf (0) = evf (2) = 10t - 2, evf (1) = 10t - 1. □

The gear graph Gn is obtained from the wheel Wn = Cn + K1 where Cn is the cycle u1u2 . . . unu1 and V (K1) = {u} by adding a vertex between every pair of adjacent vertices of the cycle Cn.

Theorem 2.3. The gear graph Gn is Total Mean cordial.

Proof. Let V (Gn) = V (Wn) ∪ {vi : 1 ≤ i ≤ n} and E (Gn) = E (Wn) ∪ {uivi, vjuj+1 : 1 ≤ i ≤ n, 1 ≤ j ≤ n} − E (Cn). Clearly p + q = 5n + 1.

Case 1. n ≡ 0 (mod 12).

Let n = 12t and t > 0. Define a map f : V (Gn) → {0, 1, 2} by f(u) = 1,

f(u5t+1) = 0, f(v12t) = 1.

Case 2. n ≡ 1 (mod 12).

Let n = 12t + 1 and t > 0. Assign the label to the vertices ui (1 ≤ i ≤ 12t), vi (1 ≤ i ≤ 12t−1) as in case 1. Then put the labels 0, 1, 2 to the vertices v12t, u12t+1, v12t+1, respectively.

Case 3. n ≡ 2 (mod 12).

Let n = 12t + 2 and t > 0. Define a map f : V (Gn) → {0, 1, 2} by f(u) = 1,

f(u9t+2) = f(v12t+2) = 2, f(u12t+2) = 0, f(u12t+1) = 1.

Case 4. n ≡ 3 (mod 12).

Let n = 12t - 9 and t > 0. Define a map f : V (Gn) → {0, 1, 2} by f(u) = 1,

f(u5t-3) = 0, f(v12t-9) = 1.

Case 5. n ≡ 4 (mod 12).

Let n = 12t − 8 and t > 0. Define a map f : V (Gn) → {0, 1, 2} by f(u) = 1,

f(u12t-8) = 1, f(v12t-8) = 2.

Case 6. n ≡ 5 (mod 12).

Let n = 12t − 7 and t > 0. Define a map f : V (Gn) → {0, 1, 2} by f(u) = 1,

f(u5t-2) = 0, f(v5t-2) = 1, f(u9t-4) = 2, f(v12t-7) = 1,

Case 7. n ≡ 6 (mod 12).

Let n = 12t − 6 and t > 0. Define a map f : V (Gn) → {0, 1, 2} by f(u) = 1,

f(u9t-4) = 2, f(v12t-6) = 2.

Case 8. n ≡ 7 (mod 12).

Let n = 12t − 5 and t > 0. Define a map f : V (Gn) → {0, 1, 2} by f(u) = 1,

f(u12t-6) = f(u12t-5) = 1, f(v12t-6) = 0, f(v12t-5) = 2.

Case 9. n ≡ 8 (mod 12

Let n = 12t − 4 and t > 0. Define a map f : V (Gn) → {0, 1, 2} by f(u) = 1,

f(u5t-1) = 0, f(v5t-1) = f(v12t-4) = 1, f(u9t-2) = 2.

Case 10. n ≡ 9 (mod 12).

Let n = 12t − 3 and t > 0. Define a map f : V (Gn) → {0, 1, 2} by f(u) = 1,

f(u9t-2) = 2, f(v12t-3) = 2.

Case 11. n ≡ 10 (mod 12).

Let n = 12t − 2 and t > 0. Define a map f : V (Gn) → {0, 1, 2} by f(u) = 1,

f(u5t) = 0, f(v12t-2) = 2.

Case 12. n ≡ 11 (mod 12).

Let n = 12t − 1 and t > 0. Define a map f : V (Gn) → {0, 1, 2} by f(u) = 1,

f(u9t) = 0, f(v12t-1) = 1.

The following table 1 shows that Gn is a Total Mean Cordial graph. □

TABLE 1.

The helm Hn is the graph obtained from a wheel by attaching a pendant edge at each vertex of the n-cycle.

Theorem 2.4. Helms Hn are Total Mean Cordial.

Proof. Let V (Hn) = {u, ui, vi : 1 ≤ i ≤ n} and E(Hn) = {uiui+1 : 1 ≤ i ≤ n − 1} ∪ {unu1} ∪ {uui, uivi : 1 ≤ i ≤ n}. Clearly the order and size of Hn are 2n + 1 and 3n respectively.

Case 1. n ≡ 0 (mod 12).

Let n = 12t and t > 0. Construct a vertex labeling f : V (Hn) → {0, 1, 2} by f(u) = 1,

and f(v5t+1) = 1.

Case 2. n ≡ 1 (mod 12).

Let n = 12t + 1 and t > 0. Define a map f : V (Hn) → {0, 1, 2} by f(u) = 1,

f(u5t+1) = 0, f(v5t+1) = 1, f(u12t) = f(u12t+1) = 1, f(v12t) = 2 and f(v12t+1) = 0.

Case 3. n ≡ 2 (mod 12).

Let n = 12t + 2 and t > 0. Define a map f : V (Hn) → {0, 1, 2} by f(u) = 1,

f(u12t+1) = f(u12t+2) = 1 and f(v12t+1) = f(v12t+2) = 2.

Case 4. n ≡ 3 (mod 12).

The Total Mean Cordial labeling of H3 is given in figure 3.

FIGURE 3.

Let n = 12t + 3 and t > 0. Define a map f : V (Hn) → {0, 1, 2} by f(u) = 1,

f(u5t+2) = 0, f(v5t+2) = 1.

Case 5. n ≡ 4 (mod 12).

Let n = 12t - 8 and t > 0. Define a map f : V (Hn) → {0, 1, 2} by f(u) = 1,

f(u12t-8) = 1 and f(v12t-8) = 2.

Case 6. n ≡ 5 (mod 12).

The Total Mean Cordial labeling of H3 is given in figure 4.

FIGURE 4.

Let n = 12t + 5 and t > 0. Define a function f : V (Hn) → {0, 1, 2} by f(u) = 1,

f(u5t+3) = 0, f(v5t+3) = 1, f(u12t+4) = f(u12t+5) = 1 and f(v12t+4) = f(v12t+5) = 2.

Case 7. n ≡ 6 (mod 12).

Let n = 12t - 6 and t > 0. Define a function f : V (Hn) → {0, 1, 2} by f(u) = 1,

Case 8. n ≡ 7 (mod 12).

Let n = 12t - 5 and t > 0. Define a function f : V (Hn) → {0, 1, 2} by f(u) = 1,

f(u12t-6) = f(u12t-5) = 1, f(v12t-6) = 2 and f(v12t-5) = 0.

Case 9. n ≡ 8 (mod 12).

Let n = 12t - 4 and t > 0. Construct a vertex labeling f : V (Hn) → {0, 1, 2} by f(u) = 1,

f(u5t-1) = 0, f(v5t-1) = 1, f(u12t-5) = f(u12t-4) = 1 and f(v12t-5) = f(v12t-4) = 2.

Case 10. n ≡ 9 (mod 12).

Let n = 12t - 3 and t > 0. Define f : V (Hn) → {0, 1, 2} by f(u) = 1,

Case 11. n ≡ 10 (mod 12).

Let n = 12t - 2 and t > 0. Define a function f : V (Hn) → {0, 1, 2} by f(u) = 1,

f(u5t) = 0, f(v5t) = 1, f(u12t-2) = 1 and f(v12t-2) = 2.

Case 12. n ≡ 11 (mod 12).

Let n = 12t - 1 and t > 0. Define a function f : V (Hn) → {0, 1, 2} by f(u) = 1,

f(u12t-2) = f(u12t-1) = 1 and f(v12t-2) = f(v12t-1) = 2.

The following table 2 shows that Hn is a Total Mean Cordial graph. □

TABLE 2.