Antimagic labelling of graphs with maximum degree $Δ(G) = n - 4$

Abstract

An antimagic labelling of a graph $G = (V,E)$ is a bijection from $E$ to $\{1,2, \ldots, |E|\}$, such that all vertex-sums are pairwise distinct, where the vertex-sum of each vertex is the sum of labels over edges incident to this vertex. A graph is said to be antimagic if it has an antimagic labelling. It has been proven that graphs $G$ with $Δ(G) \geq n - 3$ are antimagic, where $Δ(G)$ is the maximum degree of a vertex in $G$ and $n = |V|$. In this article, we extend this result to graphs with $Δ(G) = n - 4$, provided that $|E| \geq 7n$.

Figures (1)

• Figure 1: Illustration of the labelling of $G$.

Theorems & Definitions (12)

• Conjecture 1
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• Theorem 4
• Conjecture 2
• Theorem 5
• Definition 2.1
• Corollary 1