关于Perrin cordial标号：图标号理论中的一种新方法
On Perrin Cordial Labeling: A New Approach in Graph Labeling Theory
AI总结：
本文提出了受Perrin数列启发的Perrin cordial标号概念，通过单射顶点标号及模2诱导边标号定义，并探讨了其在不同图族中的存在性，为数论与图结构建立了新联系。
摘要：
在本文中，我们引入了Perrin cordial标号的概念，这是一种受Perrin数列启发的新型顶点标号方案，位于更广泛的图标号理论框架内。Perrin数由递推关系P_n = P_{n-2} + P_{n-3}定义，初始值为P_0 = 0、P_1 = 3和P_2 = 0。图G = (V, E)的Perrin cordial标号是一个单射函数f : V(G) → {P_0, P_1, ..., P_n}，其中诱导的边标号f* : E(G) → {0,1}由f*(uv) = (f(u) + f(v)) mod 2给出。如果标号为0的边数e_f(0)与标号为1的边数e_f(1)满足条件|e_f(0) - e_f(1)| ≤ 1，则称该标号为cordial。允许这种标号的图被称为Perrin cordial图。本研究通过分析各类图族的结构性质及其与所提标号方案的兼容性，探讨了Perrin cordial标号在不同图族中的存在性。我们的结果旨在丰富图标号理论，并突出数论与图结构之间的新联系。
英文摘要：
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.
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