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

TL;DR

This paper characterizes and counts equatorially balanced C4-face-magic labelings on Klein bottle grid graphs, revealing conditions for existence and providing explicit formulas for their enumeration based on grid dimensions.

Contribution

It introduces the concept of equatorially balanced C4-face-magic Klein bottle labelings and derives exact counts and existence conditions for these labelings on grid graphs embedded in the Klein bottle.

Findings

C4-face-magic Klein bottle labelings exist iff n is even.

When m is odd, all such labelings are equatorially balanced.

Number of labelings on m x 4 grid is 2^m (m-1)! τ(m).

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 $x_{i,j} + x_{i,n+1-j} = \tfrac{1}{2} S$ 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 $n \equiv 0\pmod{4}$, or $(6\cdot 2^m)(m-1)!$ if $n \equiv 2\pmod{4}$.