Sibling graphs which are $k$-distance magic

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Document Type : Original paper

Authors

Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, Indonesia

Abstract

This paper investigates $k$-distance magic labeling, for a positive integer $k$, within the framework of sibling graphs, that is, graphs that are both self-centered and antipodal. A $k$-distance magic labeling is a vertex labeling in which the sum of the labels on all vertices at distance $k$ from any given vertex is constant throughout the graph. We establish sufficient conditions for a sibling graph to admit a $k$-distance magic labeling, covering both regular and non-regular cases. Using these conditions, we show that several well-known families of regular sibling graphs of diameter $k$ are $k$-distance magic, including cylindrical grid graphs, cyclic grid graphs, $n$-dimensional hypercubes, circulant graphs, weak Bruhat graphs, and the Möbius--Kantor graph. In addition, we construct examples of irregular sibling graphs that admit a $2$-distance magic labeling.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 01 May 2026

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