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Caterpillars are Antimagic

Antoni Lozano, Mercè Mora, Carlos Seara, Joaquín Tey

Abstract

An antimagic labeling of a graph $G$ is an injection from $E(G)$ to $\{1,2,\dots,|E(G)|\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $u$ is the sum of the labels assigned to edges incident to $u$. A graph is called antimagic when it has an antimagic labeling. Hartsfield and Ringel conjectured that every simple connected graph other than $K_2$ is antimagic and the conjecture remains open even for trees. Here we prove that caterpillars are antimagic by means of an $O(n \log n)$ algorithm.

Caterpillars are Antimagic

Abstract

An antimagic labeling of a graph is an injection from to such that all vertex sums are pairwise distinct, where the vertex sum at vertex is the sum of the labels assigned to edges incident to . A graph is called antimagic when it has an antimagic labeling. Hartsfield and Ringel conjectured that every simple connected graph other than is antimagic and the conjecture remains open even for trees. Here we prove that caterpillars are antimagic by means of an algorithm.

Paper Structure

This paper contains 7 sections, 4 theorems, 20 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Construction of an Antimagic Labeling
  3. Proof of Theorem \ref{['th:caterpillars']}
  4. Proof of Correctness
  5. Proof of Efficiency
  6. Conclusions and Open Problems
  7. Acknowledgments

Key Result

Theorem \oldthetheorem

Caterpillars are antimagic. Furthermore, there exists an algorithm that, given a caterpillar $C$ of order $n$, produces an antimagic labeling for $C$ in time $O(n \log n)$.

Figures (1)

  • Figure 1: Example of a labeling of a caterpillar with a longest path of odd length. In this case, $L_0=[1,21]$ and $L_2=[22,27]$. Top, Step 1: Labeling the pathedges and legs incident to light vertices (circled vertices). Middle, Step 2: Labeling all legs but one for each heavy vertex (squared vertices). Bottom, Step 3: Labeling the last leg for each heavy vertex.

Theorems & Definitions (9)

  • Conjecture 1
  • Conjecture 2
  • Theorem \oldthetheorem
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Conjecture 3
  • Corollary 1