Antimagic face labeling of convex polytopes based on biprisms (Q2721337)

For a sphere graph \(G\), let \(F(G)\) be the set of faces of \(G\), which include the unique face of infinite area. A connected plane graph \(G= (V,E,F)\) is said to be \((a,d)\)-face antimagic if there exists positive integers \(a\), \(d\) and a bijection \(\delta: E(G)\to \{1,2,\dots,|E(G)|\}\) such that the induced mapping \(\delta^*: F(G)\to W\) is also a bijection, where \(W= \{w(f): f\in F(G)\}= \{a,a+ d,a+ 2d,\dots, a+(|F(G)|- 1)d\}\) is the set of weights of faces. The biprism \(B_n\), \(n\geq 3\), is a graph which can be defined as the Cartesian product \(P_3\times C_n\) of a graph on three vertices with a cycle on \(n\) vertices, embedded in the plane. The authors define a family of graphs of convex polytopes based on \(B_n\), and construct \((a,d)\)-face antimagic labelings for the family. The possible values of \(d\) are determined as \(d= 2,4\) or \(6\). For \(d=2\) and 4, they construct \((9n+ 3,2)\) and \((6n+ 4,4)\)-face antimagic labelings for the family of the convex polytopes.