On Liouville difference cordial labeling of some splitting graphs

Introduction

Graph labeling is a classical and well-studied branch of graph theory that has developed into a rich area of combinatorial research. Since Rosa's [1] pioneering work in 1967, the subject has attracted considerable attention because of its elegant structure and wide range of applications. Among the various labeling schemes, cordial labeling introduced by Cahit [2] occupies an important place. In a cordial labeling, the assigned labels are required to be balanced so that the corresponding edge labels differ by at most one in frequency. This notion has led to many variants, including prime cordial labeling, product cordial labeling, and difference cordial labeling and so on. Gallian's [3] survey provides a comprehensive account of the development of graph labeling. A recent extension uses the Liouville lambda function [4,5] 𝜆(𝑛) = (-1) 𝛺 (𝑛) , where 𝛺(𝑛) denotes the number of prime factors of 𝑛, counted with multiplicity. Since 𝜆(𝑛) is multiplicative and arithmetic in nature, it serves as a natural basis for defining new cordialtype labelings. Motivated by this idea, we previously introduced Liouville Difference Cordial Labeling (LDCL) and studied it for some acyclic graphs [6] . Formally, let G = (V, E) be a simple graph with vertex set V and edge set E. In this paper, we study LDCL for the splitting graphs of several standard graph families. The splitting graph [7,8,9,10] ⇒ |𝑉| = 2𝑛 and |𝐸| = 3𝑛 -3.

Let us define the vertex labeling of the graph 𝑆𝑝𝑙(𝑃 𝑛 ) by the 1 -1 function as follows: For all 1 ≤ 𝑖 ≤ 𝑛, 𝑓(𝑢 𝑖 ) = 2 3𝑖-1 and

Then for every edge in 𝐸, the induced edge function satisfies the following relations: For all 1 ≤ 𝑖 < 𝑛,

The number of edges labelled with the labels -1 and 1 are equal and it is Let the vertex labeling of the graph 𝑆𝑝𝑙(𝑃 𝑛 • 𝐾 1 ) by the 1 -1 function as follows: For all 1 ≤ 𝑖 ≤ 𝑛,

Then by the definition of Liouville Difference cordial labeling, for every edge in 𝐸, the following results holds good. For all 1 ≤ 𝑖 < 𝑛,

The number of edges labelled with the labels -1 and 1 ⌊ Let us define the vertex labeling of the graph 𝑆𝑝𝑙(𝐿 𝑛 ) by the 1 -1 function as follows: For all 1 ≤ 𝑖 ≤ 𝑛,

Then by the definition of the induced edge function, every edge in 𝐸, the following results: For all 1 ≤ 𝑖 < 𝑛,

Case (i): When 𝑛 is odd The number of edges with the labels -1 and 1 are ⌊ Let the vertex labeling of the graph 𝑆𝑝𝑙(𝑆 𝑛 ) be defined by the 1 -1 function as follows:

Then by the definition of the induced edge function, for every edge in 𝐸, the following results holds good: For all 1 < 𝑖 ≤ 𝑛,

Case (1): Suppose if 1 -2 2𝑖-3 is a prime or a product of primes where the sum of all the exponents is odd.

Then, 𝑓 * (𝑢 1 𝑢 𝑖 ) = -1 and 𝑓 * (𝑢 1

Case (2): Suppose if 1 -2 2𝑖-3 is a product of primes where the sum of all the exponents is even.

Then, 𝑓 * (𝑢 1 𝑢 𝑖 ) = 1 and 𝑓 * (𝑢 1 ′ 𝑢 𝑛-𝑖+2 ) = -1.

Case (i): When 𝑛 is odd

The number of edges labelled with the labels -1 and 1 are equal and it is Let us consider the vertex labeling of the splitting graph 𝑆𝑝𝑙(𝐵 𝑛,𝑛 ) as follows:

Then by the definition of the induced edge function, every edge in 𝐸, satisfies the following results:

Now consider for all 1 < 𝑖 ≤ 𝑛,

Case (1): Suppose if 1 -2 2𝑖-2 is a prime or a product of primes where the sum of all the exponents is odd. Then from ( 3) and ( 4), 𝑓 * (𝑢 1 𝑢 𝑖 ) = 1 and 𝑓 * (𝑣 1 ′ 𝑣 𝑖 ) = -1. Case (2): Suppose if 1 -2 2𝑖-2 is a product of primes where the sum of all the exponents is even. Then (3) and (4) implies that 𝑓 * (𝑢 1 𝑢 𝑖 ) = -1 and 𝑓 * (𝑣 1 ′ 𝑣 𝑖 ) = 1.

And consider for all 1 < 𝑖 ≤ 𝑛,

Case (1): Suppose if 1 -2 2𝑖+1 is a prime or a product of primes where the sum of all the exponents is odd.

Then, from ( 5) and ( 6), 𝑓 * (𝑣 1 𝑣 𝑖 ) = 1 and 𝑓 * (𝑢 1 ′ 𝑢 𝑖 ) = -1. Case (2): Suppose if 1 -2 2𝑖+1 is a product of primes where the sum of all the exponents is even. Then, from ( 5) and ( 6), 𝑓 * (𝑣 1 𝑣 𝑖 ) = -1 and 𝑓 * (𝑢 1 ′ 𝑢 𝑖 ) = 1. Thus, the number of edges labeled with the labels -1 and 1 are 3𝑛 -1 and 3𝑛 -2, respectively. i.e.,𝑒 𝑓 * (-1) = 3𝑛 -1 and 𝑒 𝑓 * (1) = 3𝑛 -2

It is evident that the labeled edges always satisfy the condition |𝑒 𝑓 * (-1) -𝑒 𝑓 * (1)| ≤ 1. Hence, there exists a Liouville Difference Cordial labeling in 𝑆𝑝𝑙(𝐵 𝑛,𝑛 ), for all 𝑛 ≥ 3.

Conclusion

This paper establishes that the splitting graphs of the Path, Comb, Ladder, Star, and Bistar admit Liouville Difference Cordial Labeling. The constructions presented here extend the known family of Liouville Difference Cordial graphs and demonstrate that the splitting operation preserves this property for several important graph classes. The present study suggests several directions for further research, particularly the investigation of LDCL for other graph families and graph operations.

References

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