^{1}

^{1}

^{1}

^{1}

^{2}

^{1}

^{2}

We proved that

Let

Let

Sundaram et al. [

In 2006, Sundaram et al. [

Let

Sundaram et al. [

In this paper, we determine the product cordiality and total product cordiality of the

Let

Graph

In [

The degree of a vertex

When

For

Observe that

If

If

Suppose

Suppose

If

If

The proof is thus complete.

For

Observe that

If

If

If

If

If

If

If

If

If

If

If

If

If

The proof is thus complete.

Overall, we have the following.

When

In

Suppose

The proof is thus complete.

In a product cordial labeling of

Note that if a vertex is labeled 0, then all the incident edges must be labeled 0. Suppose

The proof is thus complete.

When

From Lemma

The proof is thus complete.

When

When

In Figure

Consider

Consider

Hence, (ii) and (iii) hold.

Consider

Hence, (iv) holds.

Consider

Consider

Hence, (ii) and (iii) hold.

Consider

Hence, (iv) holds.

Consider

Consider

Hence, (ii) and (iii) hold.

Consider

Hence, (iv) holds.

Consider

Consider

Hence, (ii) and (iii) hold.

Consider

The proof is thus complete.

1, 2, 3, and so forth denote the order of vertices with

When

We prove the conclusion is right when

After each exchange,

The proof is thus complete.

Overall, we can get the following.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the anonymous referees very much for valuable suggestions, corrections, and comments, which result in a great improvement of the original paper. The work is supported by NSFC Grant no. 11371109.