Antimagic labelling of graphs with maximum degree $\mathrm{\Delta }(G)=n-4$

An antimagic labelling of a graph $G=(V,E)$ is a bijection from $E$ to $\{1,2,\dots ,\mathrm{\mid }E\mathrm{\mid }\}$, 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 $\mathrm{\Delta }(G)\ge n-3$ are antimagic, where $\mathrm{\Delta }(G)$ is the maximum degree of a vertex in $G$ and $n=\mathrm{\mid }V\mathrm{\mid }$. In this article, we extend this result to graphs with $\mathrm{\Delta }(G)=n-4$, provided that $\mathrm{\mid }E\mathrm{\mid }\ge 7n$.