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

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 $f:V(G)\to \{P_0,P_1,\dots ,P_n\}$, where the induced edge labeling $f^{\ast }:E(G)\to \{0,1\}$ is given by $f^{\ast }(uv)=(f(u)+f(v))(\mathrm{m}\mathrm{o}\mathrm{d}2)$. 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 $\mathrm{\mid }{e}_f(0)-e_f(1)\mathrm{\mid }\le 1$. 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.