On Two Conjectures Concerning $(a,d)$-Antimagic Labellings of Antiprisms

Abstract

A connected graph $G=(V,E)$ is $(a,d)$-antimagic if there exist positive integers $a,d$ and a bijection $g:E\to \{1,2,\dots ,|E|\}$ such that the induced mapping
$$f_g=\mathrm{\Sigma }\{g(u,v):(u,v)\in E(G)\}\text{is injective and}$$
$$f_g(V)=\{a,a+d,a+2d,\dots ,a+(|V|-1)d\}.$$
In this paper, we prove two conjectures of Baca concerning $(a,d)$-antimagic labelings of antiprisms