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Further results on Parity Combination Cordial Labeling

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 |ef(0) − ef(1)| ≤ 1, where ef(0) and ef(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 |ef(0) − ef(1)| ≤ 1, where ef(0) and ef(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 Cn and K1, m by unifying a vertex of Cn and a pendent vertex of K1, m. They also proved that Wn admits a parity combination cordial labeling if and only if n ≥ 4, and conjectured that for n ≥ 4, Kn is not a parity combination cordial graph. They also proved that if G is a parity combination cordial graph, then GPn 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 K2, 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 ef(0) = ef(1). Adding a new edge e will lead to |ef(0) − ef(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 K4, K5, which is proved later using Algorithms 2.5 and 2.7, as shown in Fig. 1.

Fig. 1
figure1

PCCL for graphs with order at most 5

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

figurea

Proposition 2.6 All simple graphs of order six are PCCG, as shown in Appendix 1, except K6, K5P1, K6 − e and the following graph (using Algorithm 2.5).

figureb

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.

Table 1 The upper bound for the number of edges of a graph with n vertices

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.

figurec

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 − {3, 7} and n ≥ 2) to satisfy a PCCL is given by

$$ 1+2\sum \limits_{i=2}^n\left[{2}^{sb(i)}-2\right] $$

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.

Table 2 The upper bound for the number of edges of a graph with n vertices

A PCCL of two graphs

In this section we present a PCCL of two families of graphs, the graph K2, n and the graph \( {P}_n^{(t)} \)which is the one point union of t copies of Pn.

Proposition 3.1: The graph K2, n is PCCG for n ≡ 0, 2, 3 (mod 4).

Proof. Let the set of vertices be V(K2, n) = {u1, u2; v1, v2, …, vn} and the set of edges be E(K2, n) = {u1v1, u1v2, …, u1vn; u2v1, u2v2, …, u2vn}. It’s clear that |V(K2, n)| = n + 2 and |E(K2, n)| = 2n. Then for n ≡ 0, 2, 3 (mod 4), define the labeling function f : V(K2, n) → {1, 2, …, p = n + 2} by

$$ f\left({u}_1\right)=1,f\left({u}_2\right)=2, $$

and

$$ f\left({v}_i\right)=i+2,1\le i\le n. $$

Which will lead to ef(0) = ef(1) = n depending on proposition 2.2.

Example: The graph K2, 7 is PCCG as shown in Fig. 2.

Fig. 2
figure2

A PCCL of the graph K2,7

Proposition 3.2: The graph \( {P}_n^{(t)} \), the one point union of t copies of Pn, 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

$$ f\left({v}_i^{(j)}\right)=\left\{\begin{array}{l}\kern1.8em 1,\kern4.199998em if\;i=j=1\\ {}\left(j-1\right)\left(n-1\right)+i,\kern1.08em 2\le i\le n,\kern1.08em 1\le j\le t\end{array}\right. $$

This function will lead to 0 ≤ |ef(0) − ef(1)| ≤ 1 depending on proposition 2.2.

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

Fig. 3
figure3

A PCCL of the graph P7(4)

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References

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

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    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/pdf

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    R. 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

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    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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Correspondence to Mohamed Aboshady.

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Appendix

Appendix

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

figured

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Seoud, M., Aboshady, M. Further results on Parity Combination Cordial Labeling. J Egypt Math Soc 28, 25 (2020). https://doi.org/10.1186/s42787-020-00082-8

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Math Subject Classification

  • 05C78
  • 05C85
  • 05C30