All Polyominoes are $C_4$-face-magic

Abstract

For a planar graph $G=(V,E)$ embedded in ${\mathbb{R}}^2$, let $\mathcal{F}(G)$ denote the set of faces of $G$. Then $G$ is called a \textit{$C_n$-face-magic} 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 $F\cong C_n$, the sum of all the vertex labels along $C_n$ is a constant $c$. In this paper, we prove that all polyominoes are $C_4$-face-magic.