Antimagicness of graphs with a dominating clique

A graph $G=(V,E)$ is called antimagic if there exists a bijective labelling $f:E\to \{1,2,\dots ,\mathrm{\mid }E\mathrm{\mid }\}$ such that the vertex-sums of labels over edges incident to a given vertex are all distinct. In this paper, we extend the antimagicness results over graphs with a dominating clique. We also introduce an alternative to the usual definition of antimagic graphs, called C-antimagic, allowing for the labelling to be injective in $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},\mathrm{\mid }E\mathrm{\mid }+C\}$ instead of bijective, and show that almost all graphs with a dominating clique are 3-antimagic.