Equatorially balanced C4-face-magic labelings on Klein bottle grid graphs
Stephen J. Curran, Richard M. Low, Stephen C. Locke

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 embedded in the Klein bottle, let denote the set of faces of . Then, is called a -face-magic Klein bottle graph if there exists a bijection such that for any with , the sum of all the vertex labelings along is a constant . Let for all . We call a -face-magic Klein bottle labeling on . We consider the grid graph, denoted by , embedded in the Klein bottle in the natural way. We show that for , admits a -face-magic Klein bottle labeling if and only if is even. We say that a -face-magic Klein bottle labeling on is equatorially balanced if $x_{i,j} + x_{i,n+1-j} =…
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Taxonomy
TopicsGraph Labeling and Dimension Problems
