Every signed planar graph is $5$-choosable: A short proof and refinements

Abstract

A \emph{signed graph} is a pair $\Gs$ in which $G$ is a finite simple graph and $σ:\E(G)\to\{+1,-1\}$ is a \emph{signature}. Following Máčajová--Raspaud- Škoviera and Jin--Kang--Steffen, a \emph{proper coloring} of $\Gs$ is a map $c:\V(G)\to\Z$ with $c(u)\neσ(uv)\,c(v)$ for every edge $uv$, and $\Gs$ is \emph{signed $k$-choosable} if such a coloring exists from any list assignment $L$ with $|L(v)|\ge k$. In a celebrated two-page note, Thomassen proved that every planar graph is $5$ choosable, and Jin, Kang, and Steffen subsequently extended this to signed planar graphs. Our principal contribution is a short, self-contained, and \emph{signature-blind} proof of the latter: the inductive bookkeeping inserts one factor of $σ(\cdot)$ uniformly into every constraint, so that with $σ\equiv +1$ the argument reduces verbatim to Thomassen's original. From the strengthened extension statement (\cref{thm:main}) we deduce the main result (\cref{thm:JKS}: $\chs\Gs\le 5$ for every planar signed graph), the Máčajová--Raspaud--Škoviera signed Five-Color Theorem in the symmetric palette $\Ns{2}=\{-2,-1,0,1,2\}$, the Switching Invariance Lemma, $3$-choosability of outerplanar signed graphs, $1$-defective signed $4$-choosability of planar signed graphs, a sandwich inequality relating $\chs$ to the unsigned and positive'' choice numbers, and a polynomial-time list-coloring algorithm. Voigt's planar non-$4$-choosable graph and Mirzakhani's smaller variant show the bound $5$ is best possible. We close with examples illustrating that negative edges genuinely refine unsigned phenomena, a comparison table situating our work in the literature, and several open problems.