Odd order C₄-face-magic projective grid graphs - Stephen James Curran (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2026)
For a graph G = (V, E) embedded in the projective plane, let F(G) denote the set of faces of G. Then, G is called a Cₙ-face-magic projective graph if there exists a bijection f: V(G) → 1, 2, …, |V(G)| such that for any F ∈ F(G) with F ≅ Cₙ, the sum of all the vertex labels around Cₙ is a constant S. We consider the m × n grid graph, denoted by Pm,n, embedded in the projective plane in the natural way.Let m ≥ 3 and n ≥ 3 be odd integers. It is known that the C₄-face-magic value of a C₄-face-magic labeling on Pm,n is either 2mn+1, 2mn+2, or 2mn+3. The characterization of C₄-face-magic labelings on Pm,n having C₄-face-magic value 2mn+2 is known. In this paper, we determine a category of C₄-face-magic labelings on Pm,n for which the C₄-face-magic value is either 2mn+1 or 2mn+3. It is conjectured that these are the only C₄-face-magic labelings on Pm,n having C₄-face-magic value 2mn+1 or 2mn+3.