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D-Antimagic Labelings of Oriented Star Forests

Ahmad Muchlas Abrar, Rinovia Simanjuntak

TL;DR

This work addresses the problem of $D$-antimagic labeling for oriented graphs, focusing on oriented stars and star forests. It analyzes all orientations and distance sets $D$ with $D\subseteq\{0,1,2\}$ by explicitly characterizing when a single oriented star $\overrightarrow{K_{1,n}}$ is $D$-antimagic and providing constructive labelings. Extending to star forests, it proves a necessary condition $\min(D)=0$ and, for forests of isomorphic stars, shows this is also sufficient; it then constructs a $\Pi$ orientation with explicit labeling formulas that achieve $D$-antimagic labeling for $D\in\{\{0,1\},\{0,2\},\{0,1,2\}\}$ and generalizes to arbitrary star forests. The results yield existence of appropriate orientations for all admissible $D$ on star forests and establish a concrete open problem to enumerate all orientations that are $D$-antimagic for these sets, advancing distance-based labeling in directed graphs.

Abstract

For a distance set $D$, an oriented graph $\overrightarrow{G}$ is $D$-antimagic if there exists a bijective vertex labeling such that the sum of all labels of $D$-out-neighbors is distinct for each vertex. This paper provides all orientations and all possible $D$s of a $D$-antimagic oriented star. We provide necessary and sufficient condition for $D$-antimagic oriented star forest containing isomorphic oriented stars. We show that for all possible $D$s, there exists an orientation for a star forest to admit a $D$-antimagic labeling.

D-Antimagic Labelings of Oriented Star Forests

TL;DR

This work addresses the problem of -antimagic labeling for oriented graphs, focusing on oriented stars and star forests. It analyzes all orientations and distance sets with by explicitly characterizing when a single oriented star is -antimagic and providing constructive labelings. Extending to star forests, it proves a necessary condition and, for forests of isomorphic stars, shows this is also sufficient; it then constructs a orientation with explicit labeling formulas that achieve -antimagic labeling for and generalizes to arbitrary star forests. The results yield existence of appropriate orientations for all admissible on star forests and establish a concrete open problem to enumerate all orientations that are -antimagic for these sets, advancing distance-based labeling in directed graphs.

Abstract

For a distance set , an oriented graph is -antimagic if there exists a bijective vertex labeling such that the sum of all labels of -out-neighbors is distinct for each vertex. This paper provides all orientations and all possible s of a -antimagic oriented star. We provide necessary and sufficient condition for -antimagic oriented star forest containing isomorphic oriented stars. We show that for all possible s, there exists an orientation for a star forest to admit a -antimagic labeling.
Paper Structure (5 sections, 11 theorems, 7 equations, 9 figures)

This paper contains 5 sections, 11 theorems, 7 equations, 9 figures.

Table of Contents

  1. Introduction
  2. $D$-Antimagic Oriented Stars
  3. $D$-Antimagic Oriented Stars with $max(D)=1$
  4. $D$-Antimagic Oriented Stars with $max(D)=2$
  5. $D$-Antimagic Labelings of Oriented Star Forests

Key Result

Lemma 1.1

All oriented graphs are $\{0\}$-antimagic.

Figures (9)

  • Figure 1: $\{1\}$-antimagic labelings of $\overrightarrow{K_{1,n}}$ where (a) $n = 1$ and (b) $n = 2$ and $t = 1$.
  • Figure 2: A $\{0,1\}$-antimagic labeling of a $\overrightarrow{K_{1,5}}$.
  • Figure 3: A $\{0,2\}$-antimagic labeling of a $\overrightarrow{K_{1,5}}$.
  • Figure 4: A $\{1,2\}$-antimagic labeling of the $\overrightarrow{K_{1,2}}$.
  • Figure 5: A {0,1,2}-antimagic labeling of a $\overrightarrow{K_{1,5}}$.
  • ...and 4 more figures

Theorems & Definitions (20)

  • Lemma 1.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.1
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 10 more