$E_A$-cordial labeling of graphs and its implications for $A$-antimagic labeling of trees

TL;DR

This work extends the notion of cordial labeling to $E_A$-cordial labeling for graphs with edge labels in a finite abelian group $A$, examining how the induced vertex sums influence label balance. It proves a complete characterization for when paths $P_n$ are $E_A$-cordial for cyclic $A$, and uses these results to deduce that $P_n$ is $A$-antimagic precisely when $n otormula{=2mod 4}$; cycles and paths are cordial for odd-order $A$, with a nuanced treatment for even-order groups. The paper also counters a conjecture on $A^*$-antimagic labeling of trees by showing that the weaker $A$-cordial framework does not extend as hoped in general, and provides constructive labeling schemes for $A$-antimagic paths across various group structures, including explicit constructions for $(Z_2)^m$ and methods via $R^*$-sequenceability. Overall, it advances understanding of how the algebraic structure of $A$ governs cordial and antimagic properties in trees and paths, offering explicit labeling recipes and identifying open problems for $A^*$-antimagic trees.

Abstract

If $A$ is a finite Abelian group, then a labeling $f \colon E (G) \rightarrow A$ of the edges of some graph $G$ induces a vertex labeling on $G$; the vertex $u$ receives the label $\sum_{v\in N(u)}f (v)$, where $N(u)$ is an open neighborhood of the vertex $u$. A graph $G$ is $E_A$-cordial if there is an edge-labeling such that (1) the edge label classes differ in size by at most one and (2) the induced vertex label classes differ in size by at most one. Such a labeling is called $E_A$-cordial. In the literature, so far only $E_A$-cordial labeling in cyclic groups has been studied. The corresponding problem was studied by Kaplan, Lev and Roditty. Namely, they introduced $A^*$-antimagic labeling as a generalization of antimagic labeling \cite{ref_KapLevRod}. Simply saying, for a tree of order $|A|$ the $A^*$-antimagic labeling is such $E_A$-cordial labeling that the label $0$ is prohibited on the edges. In this paper, we give necessary and sufficient conditions for paths to be $E_A$-cordial for any cyclic $A$. We also show that the conjecture for $A^*$-antimagic labeling of trees posted in \cite{ref_KapLevRod} is not true.

Figures (4)

• Figure 1: A $((\mathbb{Z}_2)^3)^*$-antimagic labeling for a tree.
• Figure 2: An $E_{\mathbb{Z}_8\oplus\mathbb{Z}_3}$-cordial labeling for $P_{24}$.
• Figure 3: An $E_{\mathbb{Z}_{24}}$-cordial labeling for $P_{24}$.
• Figure 4: A ${(\mathbb{Z}_{2})^3}$-antimagic labeling for $P_{8}$.

Theorems & Definitions (20)

• Theorem 1.1: HoveyTao
• Theorem 1.2: Cic
• Theorem 1.3: YilCah
• Theorem 1.4: LiuLiu
• Conjecture 1.5: ref_KapLevRod
• proof
• proof
• Corollary 3.3
• Corollary 3.4
• Theorem 3.5