DIFFERENCE CORDIALITY OF SOME SNAKE GRAPHS

Abstract

Let G be a (p, q) graph. Let f be a map from V (G) to {1, 2, ${\ldots}$, p}. For each edge uv, assign the label ${\mid}f(u)-f(\nu){\mid}$. f is called a difference cordial labeling if f is a one to one map and ${\mid}e_f(0)-e_f(1){\mid}{\leq}1$ where $e_f(1)$ and $e_f(0)$ denote the number of edges labeled with 1 and not labeled with 1 respectively. A graph with admits a difference cordial labeling is called a difference cordial graph. In this paper, we investigate the difference cordial labeling behavior of triangular snake, Quadrilateral snake, double triangular snake, double quadrilateral snake and alternate snakes.

1. Introduction

Let G be a (p, q) graph. In this paper we have considered only simple and undirected graph. The number of vertices of G is called the order of G, de-noted by |V (G)| and the number of edges of G is called the size of G, denoted by |E (G)|. Labeled graphs are used in several areas such as astronomy, radar, circuit design and database management [1]. The notion of difference cordial labeling has been introduced by R. Ponraj, S. Sathish Narayanan and R. Kala in [3]. In [3,4,5,6,7], difference cordial labeling behaviour of several graphs like path, cycle, complete graph, complete bipartite graph, bistar, wheel, web and some more standard graphs have been investigated. In this paper we investi-gate the difference cordial labeling behaviour of Triangular snake, Quadrilateral snake, Alternate triangular snake, Alternate quadrilateral snake. Let x be any real number. Then ⌊x⌋ stands for the largest integer less than or equal to x and ⌈x⌉ stands for the smallest integer greater than or equal to x. Terms and definitions not defined here are used in the sense of Harary [2].

 

2. Main results

Definition 2.1. Let G be a (p, q) graph. Let ƒ : V (G) → {1, 2, . . . , p} be a bijection. Foreach edge uv, assign the label |ƒ (u) − ƒ (v)|. ƒ is called a difference cordial labeling if ƒ is 1−1 and |eƒ (0) − eƒ (1)| ≤ 1 where eƒ (1) and eƒ (0) denote the number of edges labeled with 1 and not labeled with 1 respectively. A graph with a difference cordial labeling is called a difference cordial graph.

Now we investigate the difference cordial labeling behavior of some snake graphs. The triangular snake Tn is obtained from the path Pn by replacing each edge of the path by a triangle C3.

Theorem 2.2. The Triangular snake Tn is difference cordial.

Proof. Let Pn be the path u1u2 . . . un. Let V (Tn) = V (Pn)∪{vi : 1 ≤ i ≤ n − 1} and E (Tn) = E (Pn) ∪ { : 1 ≤ i ≤ n − 1}. In this graph, |V (Tn)| = 2n − 1 and |E (Tn)| = 3n − 3. For n > 4, define ƒ : V (Tn) → {1, 2, . . . , 2n − 1} by

The following table 1 shows that the labeling ƒ defined above is a difference cordial labeing of Tn for n > 4.

TABLE 1

We now display a difference cordial labeling for T2, T3 and T4 is given in figure 1.

Figure 1.

The Quadrilateral snake Qn is obtained from the path Pn by replacing each edge of the path by a cycle C4.

Theorem 2.3. All Quadrilateral snakes are difference cordial.

Proof. Let Pn be the path u1u2 . . . un. Let V (Qn) = {vi,wi : 1 ≤ i ≤ n − 1} ∪ V (Pn) and E (Qn) = E (Pn) ∪ { : 1 ≤ i ≤ n − 1}. Note that |V (Qn)| = 3n − 2 and |E Qn)| = 4n − 4. Define a map ƒ : V (Qn) → {1, 2, 3, . . . , 3n − 2} by ƒ (v1) = 3n − 3, ƒ (v2) = 3n − 2, ƒ (w1) = 3n − 4,

Here, eƒ (0) = eƒ (1) = 2n − 2. It follows that ƒ satisfies the edge condition of difference cordial graph.

Next is the alternate triangular snake. An alternate triangular snake A(Tn) is obtained from a path u1u2 . . . un by joining ui and ui+1 (alternatively) to new vertex vi. That is every alternate edge of a path is replaced by C3.

Theorem 2.4. Alternate triangular snakes are difference cordial.

Proof. Case 1. Let the first triangle be starts from u2 and the last triangle ends with un−1

In this case, and |E (A(Tn))| = 2n − 3. Define by ƒ (u1) = 1,

and In this case eƒ (0) = n − 1 and eƒ (1) = n − 2 and hence ƒ is difference cordial labeling.

Case 2. Let the first triangle be starts from u1 and the last triangle ends with un. Here and |E (A(Tn))| = 2n − 1. Define a map by

Since eƒ (0) = n − 1 and eƒ (1) = n, f is a required difference cordial labeling.

Case 3. Let the first triangle be starts from u2 and the last triangle ends with un. Note that in this case, and |E (A(Tn))| = 2n−2. Define an injective map by

Here eƒ (0) = n − 1 and eƒ (1) = n - 1. Therefore, ƒ is a difference cordial labeling.

Case 4. Let the first triangle be starts from u1 and the last triangle ends with un−1. This case is equivalent to case 3.

Now we look into alternate quadrilateral snake. An alternate quadrilateral snake A(Qn) is obtained from a path u1u2 . . . un by joining ui, ui+1 (alterna-tively) to new vertices vi, wi respectively and then joining vi and wi. That is every alternate edge of a path is replaced by a cycle C4.

Theorem 2.5. All alternate quadrilateral snakes are difference cordial.

Proof. Case 1. Let the first cycle C4 be starts from u2 and the last cycle be ends with un−1. Note that in this case, |V (A(Qn))| = 2n−2 and Define ƒ : V (A(Qn)) → {1, 2, . . . , 2n − 2} as follows:

The table 2 given below shows that ƒ is a difference cordial labeling.

TABLE 2

Case 2. Let the first cycle C4 be starts from u1 and the last cycle be ends with un. Here, |V (A(Qn))| = 2n and Define ƒ : V (A(Qn)) → {1, 2, . . . , 2n} by

In this case the following table 3 shows that ƒ is a difference cordial labeling.

TABLE 3

Case 3. Let the first cycle C4 be starts from u2 and the last cycle be ends with un. Note that |V (A(Qn))| = 2n − 1 and Define ƒ : V (A(Qn)) → {1, 2, . . . , 2n − 1} by

The following table 4 shows that ƒ is a difference cordial labeling.

TABLE 4

Case 4. Let the first cycle C4 be starts from u1 and the last cycle be ends with un−1. This case is equivalent to case 3.

Next investigation is about the irregular triangular snakes. The irregular triangular snake ITn is obtained from the path Pn : u1u2 . . . un with vertex set V (ITn) = V (Pn) → {vi : 1 ≤ i ≤ n − 2} and the edge set E (ITn) = E (Pn) ∪ { : 1 ≤ i ≤ n − 2}.

Theorem 2.6. The irregular triangular snake is difference cordial.

Proof. Clearly |V (ITn)| = 2n−2 and |E (ITn)| = 3n−5. Define ƒ : V (ITn) → {1, 2, . . . , 2n − 2} as follows:

Case 1. n is odd.

Case 2. n is even. Label the vertices ui (1 ≤ i ≤ n − 1) and vi (1 ≤ i ≤ n) as in case 1 and assign the label 2n − 2 to the vertex un. The following table 5 gives the nature of the edge condition of the above labeling ƒ. It follows that ƒ is a difference cordial labeling.

The difference cordial labeling of irregular triangular snake IT12 is given in figure 2.

TABLE 5

Figure 2.

The irregular quadrilateral snake IQn is obtained from the path Pn : u1u2 . . . , un with vertex set V (IQn) = V (Pn)∪{vi,wi : 1 ≤ i ≤ n − 2} and edge set E (IQn) = E (Pn) ∪ { : 1 ≤ i ≤ n − 2}.

Theorem 2.7. The irregular quadrilateral snake is difference cordial.

Proof. Clearly, |V (IQn)| = 3n − 4 and |E (IQn)| = 4n − 7 respectively. Define ƒ : V (IQn) → {1, 2, 3, . . . , 3n − 4} by

Since it follows that ƒ is a difference cordial labeling.

A double triangular snake DTn consists of two triangular snakes that have a common path. That is, a double triangular snake is obtained from a path u1, u2 . . . un by joining ui and ui+1 to a new vertex vi (1 ≤ i ≤ n − 1) and to a new vertex wi (1 ≤ i ≤ n − 1).

Theorem 2.8. Double triangular snake DTn is difference cordial iff n ≤ 6.

Proof. For n ≤ 6, the difference cordial labeling is given in figure 3. Conversely, suppose n > 6 and ƒ is a difference cordial labeling of the double triangular snake. Here, |V (DTn)| = 3n − 2 and |E (DTn)| = 5n − 5. We observe that the maximum value of eƒ (1) does not exceed 1 + 2 (n − 1) + 1 = 2n. Hence eƒ (0) ≥ q −2n ≥ 3n−5. Therefore, eƒ (0)−eƒ (1) ≥ n−5, a contradiction.

A double quadrilateral snake DQn consists of two triangular snakes that have a common path.

Figure 3.

Theorem 2.9. The double quadrilateral snake is difference cordial.

Proof. Let V (DQn) = {ui : 1 ≤ i ≤ n} ∪ {vi, wi, xi, yi : 1 ≤ i ≤ n − 1} and E (DQn) = : 1 ≤ i ≤ n − 1}. Clearly, |V (DQn)| = 5n − 4 and |E (DQn)| = 7n − 7. Define a map ƒ : V (DQn) → {1, 2, . . . , 5n − 4} as follows:

The following table 6 shows that ƒis a difference cordial labeling of DQn.

TABLE 6

A double alternate triangular snake DA(Tn) consists of two alternate trian-gular snakes that have a common path. That is, a double alternate triangular snake is obtained from a path u1u2 . . . un by joining ui and ui+1 (alternatively) to two new vertices vi and wi.

Theorem 2.10. Double alternate triangular snake DA(Tn)is difference cordial.

Proof. Case 1. The triangles starts from u1 and end with un. In this case, |V (DA(Tn))| = 2n and |E (DA(Tn))| = 3n − 1. Define an injective map ƒ : V (DA(Tn)) → {1, 2, . . . , 2n} by

Since ef (1) ƒ is a difference cordial labeling of DA(Tn).

Case 2. The triangles starts from u2 and end with un−1. Note that In this case, |V (DA(Tn))| = 2n − 2 and |E (DA(Tn))| = 3n − 5. Label the vertices vi and as in case 1 and define ƒ (ui) = 2i − 3, 2 ≤ i ≤ n − 1, ƒ (u1) = 2n − 3 and ƒ (un) = 2n − 2. Since ƒ is a difference cordial labeling of DA(Tn).

Case 3. The triangles starts from u2 and end with un. It is clear that in this case, |V (DA(Tn))| = 2n − 1 and |E (DA(Tn))| = 3n − 3. Label the vertices vi and as in case 1 and label ui (2 ≤ i ≤ n − 1) as in case 2 and define ƒ (u1) = 2n − 1. Since ƒ is a difference cordial labeling of DA(Tn).

Case 4. The triangles starts from u1 and end with un−1. This case is equivalent to case 3.

Finally, we look into the graph double alternate quadrilateral snake. A double alternate quadrilateral snake DA(Qn) consists of two alternate quadrilateral snakes that have a common path. That is, it is obtained from a path u1u2 . . . un by joining ui and un+1 (alternatively) to new vertices vi, xi and wi, yi respectively and adding the edges viwi and xiyi.

Theorem 2.11. All double alternate quadrilateral snakes are difference cordial.

Proof. Case 1 The squares starts from u1 and end with un. In this case, |V (DA(Qn))| = 3n and |E (DA(Qn))| = 4n−1. Define a map ƒ : V (DA(Qn)) → {1, 2, . . . , 3n} by

Since eƒ (1) = 2n and eƒ (0) = 2n − 1, ƒ is a difference cordial labeling of DA(Qn).

Case 2. The squares starts from u2 and end with un−1. In this case, |V (DA(Qn))| = 3n−4 and ||E (DA(Qn))| = 4n−7. Label the vertices vi, wi, xi, yi as in case 1 and define ƒ (ui) = 3i − 4, 2 ≤ i ≤ n, and ƒ (u1) = 3n − 5. Since eƒ (1) = 2n−3 and eƒ (0) = 2n−4, ƒ is a difference cordial labeling of DA(Qn).

Case 3. The square starts from u2 and end with un. In this case, |V (DA(Qn))| = 3n−2 and |E (DA(Qn))| = 4n−4. Label the vertices vi, wi, xi, yi as in case 1 and label ui (2 ≤ i ≤ n) as in case 2 and define ƒ (u1) = 3n − 2. Since eƒ (1) = ef (0) = 2n − 2, ƒ is a difference cordial labeling of DA(Qn).

Case 4. The square starts from u1 and end with un-1. This case is equivalent to case 3.