\(C_4\) -Face-Magic Labelings on Even Order Projective Grid Graphs

For a graph \(G = (V, E)\) embedded in the projective plane, let \(\mathcal {F}(G)\) denote the set of faces of G. Then, G is called a \(C_n\) -face-magic projective graph if there exists a bijection \(f: V(G) \rightarrow \{1, 2, \dots , |V(G)|\}\) such that for any \(F \in \mathcal {F}(G)\) with \(F \cong C_n\) , the sum of all the vertex labelings along \(C_n\) is a constant S. Let \(x_v =f(v)\) for all \(v\in V(G)\) . We call \(\{x_v : v\in V(G)\}\) a \(C_n\) -face-magic projective labeling on G. We consider the \(m \times n\) grid graph, denoted by \(\mathcal {P}_{m,n}\) , embedded in the projective plane in the natural way. It is known that, for \(m,n\ge 2\) , \(\mathcal {P}_{m,n}\) admits a \(C_4\) -face-magic projective labeling if and only if m and n have the same parity. When m and n are even, a \(C_4\) -face-magic projective labeling on \(\mathcal {P}_{m,n}\) has \(C_4\) -face-magic value \(2mn+2\) . We show that there are 144 distinct \(C_4\) -face-magic projective labelings on the \(4\times 4\) projective grid graph \(\mathcal {P}_{4,4}\) (up to symmetries on the projective plane).