1. Introduction and Definitions

In1987, Cahit[1] introduced the notion of cordial labeling. Sundaram, Ponraj and Somasundar-am[6] introduced the notions of product cordial labeling and total product cordial labeling in 2006. As for the detailed results on cordial graph, product cordial graph and total product cordial graph, the readers can be referred to[2]. Lee and Wang[4] defined pyramid graph, and proved that pyramid graph is graceful. Most graph labeling methods trace their original one introduced by Rosa[5] in 1967, or one given by Graham and Sloane[3] in 1980. In this paper, graph decomposition method was introduced, it is proved that pyramid graphs are cordial and total product cordial.

Definition 1.1.Cordial Graph: a cordial labeling of a graph with vertex set is a function from to {0, 1} such that if each edge assigned the label , the number of vertices labeled with 1 and the number of vertices labeled with 0 differ by at most 1, the number of edges labeled with 1 and the number of edges labeled with 0 differ by at most 1. A graph is called cordial graph if it admits a cordial labeling.

Definition 1.2. total product cordial graph: a total product cordial labeling of a graph with vertex set is a function from to {0,1} such that if each edge is assigned the label , the number of vertices and edges labeled with 1 and the number of vertices and edges labeled with 0 differ by at most 1. A graph is called total product cordial graph if it admits a total product cordial labeling.

Definition 1.3. pyramid graph: a pyramid graph obtained by arranging vertices into a finite number of lines with vertices in the th line and every line the th vertex in that line is joined to the th vertex and the (+1)th vertex of the next line. A pyramid graph that has vertices is denoted by is illustrated in Fig.1.

2. Results

First, we let ,

Theorem 2.1. is cordial graph.

In the follow discussion, is divided into some subgraphs, and defined the vertex labelings in these subgraphs, to discuss .

Proof the vertices of are divided into groups every four lines successively and the vertices in the rest lines (if exists) form one group, in each group, there are odd number of vertices in the first line and the third line, there are even number of vertices in the second line and the fourth line. The rules of vertices in each group are defined as follows:

1. the vertex labelings in the same line are alternating

2. the first vertex labeling in the first line and the fourth line are ;

3. the first vertex labeling in the second line and the third line are .

Based on the previous rules, in every group, the number of vertices labeled with 0 is one more than the number of vertices labeled with 1 in the first line, the number of vertices labeled with 1 is one more than the number of vertices labeled with 0 in the third line, the number of vertices labeled with 0 is the same as the number of vertices labeled with 1 in the second and fourth lines, therefore

when

when there is only one line in the last group, so

Figure 1. pyramid

when, there are only two lines in the last group, so

when, there are three lines in the last group,

Except the vertices in the last line, each vertex in other lines is adjacent to two vertices in the next line in, based on the previous rules and definition 1.1, it is obtained that edge labeled with 0 and 1 which are got between each vertex in each line and adjacent vertices in the next line appear in pairs, so , therefore, is cordial.

Theorem 2.2. is total product cordial graph.

In the follow discussion, is divided into two part, one is , another is the last two lines in , used the vertex labelings in the last two lines and the total product cordial property of , to discuss .

Lemma 2.1. if is even, then is total product cordial graph.

Proof First, the rules are defined as follows:

1. the vertex labelings of the first lines in are the vertex labelings in ;

2. in，the vertex labelings in the odd lines are

3. when , successive six graphs from one group;

4. in，the vertex labelings in the last line are the number of vertices labeled withisand the number of vertices labeled with is in the last line;

5. in ，the vertex labelings in the last line are the number of vertices labeled with is and the number of vertices labeled with is in the last line;

6. in，the vertex labelings in the last line are the number of vertices labeled withis and the number of vertices labeled with is in the last line；

7. in，the vertex labelings in the last line are the number of vertices labeled with is and the number of vertices labeled with is in the last line；

8. in，the vertex labelings in the last line are the number of vertices labeled with is and the number of vertices labeled with is in the last line；

9. in，the vertex labelings in the last line are the number of vertices labeled with is and the number of vertices labeled with is in the last line.

The vertex labelings in are illustrated in Fig.2, according to the rule, we obtain that are total product cordial too.

Figure 2. The vertex labelings in

of the first group are discussed as follows. The followings are got by calculation:

in

in

in

so are total product cordial,

Suppose, in the group are total product cordial, the labelings of every graph in the th group are discussed.

In，according to rule 4, the number of vertices labeled with is and the number of vertices labeled with is in the last line; according to rule 2, the number of vertices labeled with is and the number of vertices labeled with is in the second line from the bottom. The numbers of edges labeled with and with in the last two lines are discussed, because the vertex labelings are alternating and 1in in the second line from the bottom, the vertex labelings in the third line from the bottom are the vertex labelings in the last line in in edges that between the vertices in the third line from the bottom and the vertices in the second line from the bottom, the number of edges labeled with the number of edges labeled with ; in edges that between the vertices in the second line from the bottom and the vertices in the last line, the number of edges labeled with the number of edges labeled with is , so the number of vertices and edges labeled with in the last two lines of is, the number of vertices and edges labeled with is ，because so

Inaccording to rule 5, the number of vertices labeled with and the number of vertices labeled with in the last line; according to rule 2, the number of vertices labeled with and the number of vertices labeled with in the second line from the bottom, the vertex labelings in the third line from the bottom are the vertex labelings in the last line in in edges that between the vertices in the third line from the bottom and the vertices in the second line from the bottom, the number of edges labeled withis, the number of edges labeled withis,similarly, the number of the edges in the last two lines labeled with the number of edges in the last two lines labeled with isso the number of vertices and edges labeled with in the last two lines of the number of vertices and edges labeled with in the last two lines is because in , so

The discussions of labeling number in the next part are similar to the above ones, therefore the number of the labelings of vertices and edges in the last two lines are given as follows.

In，according to rule 2 and 6, the number of vertices labeled with ，the number of vertices labeled with the number of edges labeled with the number of edges labeled with

In ，according to rule 2 and 7, the number of vertices labeled with the number of vertices labeled with the number of edges labeled with the number of edges labeled with

In according to rule 2 and 8, the number of vertices labeled with the number of vertices labeled with the number of edges labeled with the number of edges labeled with

Inaccording to rule 2 and 9, the number of vertices labeled with the number of vertices labeled withthe number of edges labeled with the number of edges labeled with

Lemma 2.2. if is total product cordial graph.

In the proof of lemma 2.2, the discussion is similar to that in lemma2.1, therefore only the rules and some of results in different cases are given.

Proof First, the rules are given as follows:

1. the vertex labelings in the first are the vertex labelings in

2. inthe vertex labelings in the even lines are

3. when successive six graphs from one group;

4. inthe vertex labelings in the last line are the number of vertices labeled with and the number of vertices labeled withis in the last line;

5. in the vertex labelings in the last line are the number of vertices labeled with and the number of vertices labeled with is in the last line;

6. inthe vertex labelings in the last line are the number of vertices labeled with and the number of vertices labeled with is in the last line;

7. in the vertex labelings in the last line are the number of vertices labeled with and the number of vertices labeled with in the last line;

8. inthe vertex labelings in the last line are the number of vertices labeled with and the number of vertices labeled with is in the last line;

9.inthe vertex labelings in the last line are the number of vertices labeled with and the number of vertices labeled with is in the last line.

The vertex labelings in are illustrated in Fig.3. According to rule 1， are total product cordial .

Figure 3. The vertex labelings in

of the first group are discussed as follows. The followings are got by calculation:

in

in

in.

In

in in

Overall, is total product cordial.

ACKNOWLEDGEMENTS

The authors would like to thank the anonymous referees very much for valuable suggestions, corrections and comments, which results in a great improvement in the original manuscript.