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

Abstract

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 $m⩾3$ and $n⩾3$ 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 $2mn+3$.

22 pages, 3 figures