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# Further results on Parity Combination Cordial Labeling

*Journal of the Egyptian Mathematical Society*
**volume 28**, Article number: 25 (2020)

## Abstract

Let *G* be a (*p*, *q*)-graph. Let *f* be an injective mapping from *V*(*G*) to {1, 2, …, *p*}. For each edge *xy*, assign the label \( \left(\genfrac{}{}{0pt}{}{x}{y}\right) \) or \( \left(\genfrac{}{}{0pt}{}{y}{x}\right) \) according as *x* > *y* or *y* > *x*. Call *f* a parity combination cordial labeling if |*e*_{f}(0) − *e*_{f}(1)| ≤ 1, where *e*_{f}(0) and *e*_{f}(1) denote the number of edges labeled with an even number and an odd number, respectively. In this paper we make a survey on all graphs of order at most six and find out whether they satisfy a parity combination cordial labeling or not and get an upper bound for the number of edges *q* of any graph to satisfy this condition and describe the parity combination cordial labeling for two families of graphs.

## Introduction

In this paper we will deal with finite simple undirected graphs. By *G*(*V*, *E*) we mean a graph with *p* vertices and *q* edges, where *p* = |*V*| and *q* = |*E*|. We follow Harary [1] for standard terminology and notations, and see Gallian [2] for more details on graph labelings.

**Definition 1.1** [3] For a graph *G*(*p*, *q*), let *f* be an injective mapping from *V*(*G*) to {1, 2, …, *p*}. For each edge *xy*, assign the label \( \left(\genfrac{}{}{0pt}{}{x}{y}\right) \) or \( \left(\genfrac{}{}{0pt}{}{y}{x}\right) \) according as *x* > *y* or *y* > *x*. Call *f* a parity combination cordial labeling if |*e*_{f}(0) − *e*_{f}(1)| ≤ 1, where *e*_{f}(0) and *e*_{f}(1) denote the number of edges labeled with an even number and an odd number, respectively. A graph with a parity combination cordial labeling is called a parity combination cordial graph (PCCG).

Ponraj, Sathish Narayanan, and Ramasamy [3, 4] proved that the following are parity combination cordial graphs: paths, cycles, stars, triangular snakes, alternate triangular snakes, olive trees, combs, crowns, fans, umbrellas, \( {\boldsymbol{P}}_{\boldsymbol{n}}^{\mathbf{2}} \), helms, dragons, bistars, butterfly graphs, and graphs obtained from *C*_{n} and *K*_{1, m} by unifying a vertex of *C*_{n} and a pendent vertex of *K*_{1, m}. They also proved that *W*_{n} admits a parity combination cordial labeling if and only if *n* ≥ 4, and conjectured that for *n* ≥ 4, *K*_{n} is not a parity combination cordial graph. They also proved that if *G* is a parity combination cordial graph, then ** G** ∪

*P*_{n}is also parity combination cordial if

*n*≠ 2, 4.

In this paper we try to present some further results, we give the parity combination cordial labeling (PCCL) of all graphs of order at most six, make an algorithm that identify whether any graph of order *p* and size *q* can be a PCCG or not, give an upper bound to the number of edges of any graph which satisfy this condition and finally describe the PCCL function of the two graphs *K*_{2, n} and \( {P}_n^{(t)} \).

### General results

**Proposition 2.1** For a simple graph *G* with *q*-edges, if *G* is PCCL and *q* is even then *G* ± *e* is PCCG.

**Proof** Since *G* is PCCL and *q* is even, then *e*_{f}(0) = *e*_{f}(1). Adding a new edge *e* will lead to |*e*_{f}(0) − *e*_{f}(1)| = 1, satisfying the PCCL condition.

**Proposition 2.2** [2] \( \left(\genfrac{}{}{0pt}{}{n}{2}\right) \) is even if *n* ≡ 0, 1 (mod 4) and odd if *n* ≡ 2, 3 (mod 4).

**Proposition 2.3**\( \left(\genfrac{}{}{0pt}{}{n}{3}\right) \) is even if *n* ≡ 0, 1, 2 (mod 4) and odd if *n* ≡ 3 (mod 4).

**Proof** For *n* ≡ 1 (mod 4), we have three cases.

Case 1: \( n=12t+1\Longrightarrow \left(\genfrac{}{}{0pt}{}{n}{3}\right)=\frac{\left(12t+1\right)(12t)\left(12t-1\right)}{6}=2t\left(12t+1\right)\left(12t-1\right)\Longrightarrow \) even.

Case 2: \( n=12t+5\Longrightarrow \left(\genfrac{}{}{0pt}{}{n}{3}\right)=\frac{\left(12t+5\right)\left(12t+4\right)\left(12t+3\right)}{6}=2\left(12t+5\right)\left(3t+1\right)\left(4t+1\right)\Longrightarrow \) even.

Case 3: \( n=12t+9\Longrightarrow \left(\genfrac{}{}{0pt}{}{n}{3}\right)=\frac{\left(12t+9\right)\left(12t+8\right)\left(12t+7\right)}{6}=2\left(4t+3\right)\left(3t+2\right)\left(12t+7\right)\Longrightarrow \) even.

Similarly for *n* ≡ 0, 2 (mod 4).

For *n* ≡ 3 (mod 4), we have also three cases.

Case 1: \( n=12t+3\Longrightarrow \left(\genfrac{}{}{0pt}{}{n}{3}\right)=\frac{\left(12t+3\right)\left(12t+2\right)\left(12t+1\right)}{6}=\left(4t+1\right)\left(6t+1\right)\left(12t+1\right)\Longrightarrow \) odd.

Case 2: \( n=12t+7\Longrightarrow \left(\genfrac{}{}{0pt}{}{n}{3}\right)=\frac{\left(12t+7\right)\left(12t+6\right)\left(12t+5\right)}{6}=\left(12t+7\right)\left(2t+1\right)\left(12t+5\right)\Longrightarrow \) odd.

Case 3: \( n=12t+11\Longrightarrow \left(\genfrac{}{}{0pt}{}{n}{3}\right)=\frac{\left(12t+11\right)\left(12t+10\right)\left(12t+9\right)}{6}=\left(12t+11\right)\left(6t+5\right)\left(4t+3\right)\Longrightarrow \) odd.

**Proposition 2.4** All simple connected graphs of order at most five are PCCG except *K*_{4}, *K*_{5}, which is proved later using Algorithms 2.5 and 2.7, as shown in Fig. 1.

**Algorithm 2.5** We made an algorithm to test any graph whether it is PCCG or not.

**Proposition 2.6** All simple graphs of order six are PCCG, as shown in Appendix 1, except *K*_{6}, *K*_{5} ∪ *P*_{1}, *K*_{6} − *e* and the following graph (using Algorithm 2.5).

The upper bound for the number of edges *q* of any graph with *n* ≤ 100 vertices to satisfy a PCCL is computed using the following algorithm and shown in Table 1.

**Algorithm 2.7** In this algorithm we count the number of even entries and odd entries that are greater than one in the Pascal’s triangle and compute twice the smaller number plus one.

Based on the upper bound listed in Table 1, the following conjecture gives an upper bound for the number of edges of any graph to be PCCG.

**Conjecture 2.8** The upper bound for the number of edges of any graph with *n* vertices (∀*n* ∈ *ℕ* − {3, 7} and *n* ≥ 2) to satisfy a PCCL is given by

where *sb*(*i*) is number of times the digit 1 occurs in the binary representation of *i*.

In Table 2 we compare the upper bound calculated from Algorithm 2.7 and that calculated from Conjecture 2.8 for *n* = 100 vertices and found that they match in all cases except for *n* = 3 and *n* = 7.

### A PCCL of two graphs

In this section we present a PCCL of two families of graphs, the graph *K*_{2, n} and the graph \( {P}_n^{(t)} \)which is the one point union of *t* copies of *P*_{n}.

**Proposition 3.1:** The graph *K*_{2, n} is PCCG for *n* ≡ 0, 2, 3 (mod 4).

**Proof.** Let the set of vertices be *V*(*K*_{2, n}) = {*u*_{1}, *u*_{2}; *v*_{1}, *v*_{2}, …, *v*_{n}} and the set of edges be *E*(*K*_{2, n}) = {*u*_{1}*v*_{1}, *u*_{1}*v*_{2}, …, *u*_{1}*v*_{n}; *u*_{2}*v*_{1}, *u*_{2}*v*_{2}, …, *u*_{2}*v*_{n}}. It’s clear that |*V*(*K*_{2, n})| = *n* + 2 and |*E*(*K*_{2, n})| = 2*n*. Then for *n* ≡ 0, 2, 3 (mod 4), define the labeling function *f* : *V*(*K*_{2, n}) → {1, 2, …, *p* = *n* + 2} by

and

Which will lead to *e*_{f}(0) = *e*_{f}(1) = *n* depending on proposition 2.2.

Example: The graph *K*_{2, 7} is PCCG as shown in Fig. 2.

**Proposition 3.2:** The graph \( {P}_n^{(t)} \), the one point union of *t* copies of *P*_{n}, is PCCG.

**Proof.** Let the set of vertices be \( V\left({P}_n^{(t)}\right)=\left\{{v}_1^{(1)},{v}_2^{(1)},\dots, {v}_n^{(1)};{v}_2^{(2)},\dots, {v}_n^{(2)};\dots; {v}_2^{(t)},\dots, {v}_n^{(t)}\right\} \). It’s clear that \( \left|V\left({P}_n^{(t)}\right)\right|=\left(n-1\right)t+1 \) and \( \left|E\left({P}_n^{(t)}\right)\right|=\left(n-1\right)t \). Then define the labeling function \( f:V\left({P}_n^{(t)}\right)\to \left\{1,2,\dots, \left(n-1\right)t+1\right\} \) by

This function will lead to 0 ≤ |*e*_{f}(0) − *e*_{f}(1)| ≤ 1 depending on proposition 2.2.

Example: The graph \( {P}_7^{(4)} \) is PCCG as shown in Fig. 3.

## Availability of data and materials

All the data in the manuscript are public.

## References

F. Harary: Graph Theory. Addison-Wesley, Reading, MA (1969). https://shyam.nitk.ac.in/Books/GT%20Books/Harary.pdf

J.A. Gallian: A Dynamic Survey of Graph Labeling. The Elec. J. of Comb..

**21th**Ed. (2018). https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6/pdfR. Ponraj, S. Sathish Narayanan and A.M.S. Ramasamy: Parity Combination Cordial Labeling of Graphs, Jordan J. of Math. and Stat.. 8(4), (2015) 293–308. https://www.researchgate.net/publication/299031186_Parity_combination_cordial_labeling_of_graphs

R. Ponraj, Rajpal Singh and S. Sathish Narayanan: On parity combination cordial graphs, Palestine Journal of Mathematics. 6(1), (2017) 211–218. http://pjm.ppu.edu/sites/default/files/papers/PJM_MAR_2017_25.pdf

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## Appendix

### Appendix

### Appendix 1: All graphs of order six according to Harary [2] that satisfy a PCCL

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