Equatorially balanced C4-face-magic labelings on Klein bottle grid graphs

Abstract

For a graph $G=(V,E)$ embedded in the Klein bottle, let $\mathcal{F}(G)$ denote the set of faces of $G$. Then, $G$ is called a $C_k$-face-magic Klein bottle 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 $F\cong C_k$, the sum of all the vertex labelings along $C_k$ is a constant $S$. Let $x_v=f(v)$ for all $v\in V(G)$. We call $\{x_v:v\in V(G)\}$ a $C_k$-face-magic Klein bottle labeling on $G$. We consider the $m\times n$ grid graph, denoted by ${\mathcal{K}}_{m,n}$, embedded in the Klein bottle in the natural way. We show that for $m,n\ge 2$, ${\mathcal{K}}_{m,n}$ admits a $C_4$-face-magic Klein bottle labeling if and only if $n$ is even. We say that a $C_4$-face-magic Klein bottle labeling $\{x_{i,j}:(i,j)\in V({\mathcal{K}}_{m,n})\}$ on ${\mathcal{K}}_{m,n}$ is equatorially balanced if $x_{i,j}+x_{i,n+1-j}=\frac{1}{2}S$ for all $(i,j)\in V({\mathcal{K}}_{m,n})$. We show that when $m$ is odd, a $C_4$-face-magic Klein bottle labeling on ${\mathcal{K}}_{m,n}$ must be equatorially balanced. Also when $m$ is odd, we show that (up to symmetries on the Klein bottle) the number of $C_4$-face-magic Klein bottle labelings on the $m\times 4$ Klein bottle grid graph is $2^m(m-1)!\tau (m)$, where $\tau (m)$ is the number of positive divisors of $m$. Furthermore, let $m\ge 3$ be an odd integer and $n\ge 6$ be an even integer. Then, the minimum number of distinct $C_4$-face-magic Klein bottle labelings $X$ on ${\mathcal{K}}_{m,n}$ (up to symmetries on a Klein bottle) is either $(5\cdot 2^m)(m-1)!$ if $n\equiv 0(\mathrm{mod}4)$, or $(6\cdot 2^m)(m-1)!$ if $n\equiv 2(\mathrm{mod}4)$.

35 pages, 19 figures