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Trees whose even-degree vertices induce a path are antimagic

Antoni Lozano, Mercè Mora, Carlos Seara, Joaquín Tey

Abstract

An antimagic labeling a connected graph $G$ is a bijection from the set of edges $E(G)$ to $\{1,2,\dots,|E(G)|\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $v$ is the sum of the labels assigned to edges incident to $v$. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every simple connected graph other than $K_2$ is antimagic; however, the conjecture remains open, even for trees. In this note we prove that trees whose vertices of even degree induce a path are antimagic, extending a result given by Liang, Wong, and Zhu [Discrete Math. 331 (2014) 9--14].

Trees whose even-degree vertices induce a path are antimagic

Abstract

An antimagic labeling a connected graph is a bijection from the set of edges 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 if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every simple connected graph other than is antimagic; however, the conjecture remains open, even for trees. In this note we prove that trees whose vertices of even degree induce a path are antimagic, extending a result given by Liang, Wong, and Zhu [Discrete Math. 331 (2014) 9--14].

Paper Structure

This paper contains 2 sections, 4 theorems, 8 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Constructing an antimagic labeling

Key Result

Theorem \oldthetheorem

LWZ If $T$ is a tree such that $V_{even}(T)$ induces a path and $\vert V_{even}(T)\vert$ is odd, then $T$ is antimagic.

Figures (3)

  • Figure 1: Labeling of $T_1$ for $p=5$ and $m=21$; (a) before the swaps, and (b) after the swaps. The shadowed label at each vertex is the vertex sum modulo $23$. Squared vertices have the same label.
  • Figure 2: Labeling of $T_1$ for $p=4$ and $m=21$; (a) before the swap, and (b) after the swap. The shadowed label at each vertex is the vertex sum modulo $23$. Squared vertices have the same label.
  • Figure 4: An antimagic labeling of a tree with $m=21$. Thicker edges correspond to the forest $T_2$ and are labeled in Step II (in this example, the forest $T_2$ has two nontrivial components). The shadowed label at each vertex is the vertex sum modulo $23$. Squared vertices have the same label, but different vertex sums.

Theorems & Definitions (5)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Corollary 1