On complementary consecutive labelings of octahedron (Q2761068)

A consecutive vertex labeling of a plane graph \(G\) of order \(n\) is a bijection \(w\) between \(V(G)\) and \(\{1,2,\ldots ,n\}\) such that \(\{w(f): \;f\) is a face of \(G\}\) is a set of consecutive integers, where \(w(f)\) is \(\sum _{v\in \partial f}w(v)\). Consecutive edge labeling is defined analogously. It is proved that the octahedron \(O\) (platonic solid) has just two nonisomorphic consecutive vertex labelings. Namely, if \(V(O)=\{1,2,\ldots ,6\}\), \(456\) is the outer face, and the three nonedges are \(42, 53\), and \(61\), then the two labelings \(w(1)w(2)\ldots w(6)\) are: \(124365\) and \(241536\). An algorithm is described that for \(O\) constructs to a given consecutive vertex labeling \(w_1\) a complementary consecutive edge labeling \(w_2\), which means that \(w_1(f)+w_2(f)\) is a constant independent of the face \(f\).