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.

Disclosure

“(5) Explore similar questions for Cn -face-magic graphs for n ≥ 5. 6. Tool and computational resource disclosure An initial proof of Conjecture 1, including versions of Lemmas 2 and 3, was generated by OpenAI’s GPT- 5.6 Sol. For this proof search, we adapted the prompt that led to a purported proof of the longstanding Cycle Double Cover Conjecture. The full original prompt for the CDCC is available at [6]. The authors independently verified al”

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Classification

Drafting a complete proof for author revision

Multiplier

9

Verified

2026-08-23T05:06:34.158223Z

Structural counts