On Perrin Cordial Labeling: A New Approach in Graph Labeling Theory

TL;DR

This work introduces Perrin cordial labeling, a vertex-labeling scheme based on the Perrin sequence where $f^*(uv)=(f(u)+f(v)) \bmod 2$ and the goal is $|e_f(0)-e_f(1)|\le 1$. It develops parity-based constructions and proves existence results for numerous graph families, including paths, cycles, complete graphs, complete bipartite graphs, stars, wheels, triangular snakes, friendship graphs, and jellyfish graphs. The results yield precise conditions under which each family admits Perrin cordial labeling (e.g., all paths; $C_n$ when $n \not\equiv 2 \pmod 4$; finite sets for $K_n$; varied but explicit criteria for $K_{m,n}$, $S_n$, $W_n$, TS$_n$, $F_n$, and $J_{m_1,m_2}$). This work strengthens the connection between number theory and graph labeling and opens avenues for algorithmic labelings and exploration with other numerical sequences.

Abstract

In this paper, we introduce the concept of \emph{Perrin cordial labeling}, a novel vertex labeling scheme inspired by the Perrin number sequence and situated within the broader framework of graph labeling theory. The Perrin numbers are defined recursively by the relation \( P_n = P_{n-2} + P_{n-3} \), with initial values \( P_0 = 0 \), \( P_1 = 3 \), and \( P_2 = 0 \). A Perrin cordial labeling of a graph \( G = (V, E) \) is an injective function \( f : V(G) \rightarrow \{P_0, P_1, \dots, P_n\} \), where the induced edge labeling \( f^* : E(G) \rightarrow \{0,1\} \) is given by \( f^*(uv) = (f(u) + f(v)) \pmod 2 \). The labeling is said to be cordial if the number of edges labeled \( 0 \), denoted \( e_f(0) \), and the number labeled \( 1 \), denoted \( e_f(1) \), satisfy the condition \( |e_f(0) - e_f(1)| \leq 1 \). A graph that admits such a labeling is called a \emph{Perrin cordial graph}. This study investigates the existence of Perrin cordial labelings in various families of graphs by analyzing their structural properties and compatibility with the proposed labeling scheme. Our results aim to enrich the theory of graph labelings and highlight a new connection between number theory and graph structures.

Figures (5)

• Figure 1: Perrin cordial labeling for $W_{14}$ graph
• Figure 2: Triangular blades have only two possible edge compositions in cordial labeling
• Figure 3: Perrin cordial labeling of ${\rm TS}_{4}$ graph
• Figure 4: Perrin cordial labeling for $F_4$ graph
• Figure 5: Perrin cordial labeling for $J_{7,7}$

Theorems & Definitions (16)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Corollary 2.7