Face-magic labelings of some gridded graphs

Document Type : Original paper

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University of Minnesota Duluth, Duluth, MN, USA

Abstract

A type $(1,1,1)$ face-magic labeling of a planar graph $G=(V,E,F)$  is a bijection from $V\cup E \cup F$ to the set of labels $\{1,2,\dots,|V|+|E|+|F|\}$ such that the weight of every $n$-sided face of $G$ is equal to the same fixed constant. The weight of a face $\mathcal{F} \in F$ is equal to the sum of the labels of the vertices, edges, and face that determine $\mathcal{F}$. It is known that the grid graph $P_m \square P_n$ admits a type $(1,1,1)$ face-magic labeling, but the proof in the literature is quite lengthy. We give a simple proof of this result and show two more infinite families of gridded graphs admit type $(1,1,1)$ face-magic labelings.

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References

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Communications in Combinatorics and Optimization
Volume 8, Issue 3
September 2023
Pages 595-601

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