Odd order $C_4$-face-magic $m \times n$ projective grid graphs having $C_4$-face-magic value $2mn+1$ or $2mn+3$

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 $f: V(G) \to \{1, 2, \dots, |V(G)|\}$ such that for any $F \in \mathcal{F}(G)$ with $F \cong C_n$, the sum of all the vertex labels around $C_n$ is a constant $S$. We consider the $m \times n$ grid graph, denoted by $\mathcal{P}_{m,n}$, embedded in the projective plane in the natural way. Let $m \geqslant 3$ and $n \geqslant 3$ be odd integers. It is known that the $C_4$-face-magic value of a $C_4$-face-magic labeling on $\mathcal{P}_{m,n}$ is either $2mn+1$, $2mn+2$, or $2mn+3$. The characterization of $C_4$-face-magic labelings on $\mathcal{P}_{m,n}$ having $C_4$-face-magic value $2mn+2$ is known. In this paper, we determine a category of $C_4$-face-magic labelings on $\mathcal{P}_{m,n}$ for which the $C_4$-face-magic value is either $2mn+1$ or $2mn+3$. It is conjectured that these are the only $C_4$-face-magic labeling on $\mathcal{P}_{m,n}$ having $C_4$-face-magic value $2mn+1$ or $2mn+3$.

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