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Complement Submodular Information Measures for Balanced and Robust Data Selection On The Morse Ensemble Polynomial Of Simplicial Complexes A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring An identity for second Eulerian numbers via lattice-point counting $t$-tone edge coloring of graphs Constructing Maximal Bumpless Pipedreams for Double Grothendieck Polynomials Mubayi's Polynomial-Ideal Conjecture and Cover-Ideal Turán Methods Implicit Binarization via Complex Phase Dynamics in Combinatorial Optimization The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function Universality theorems for generalized splines Framing Triangulations for Arbitrary Integer Flow Polytopes On the Common Generalization of Gentle Algebras and Framed Directed Acyclic Graphs The complexity of frugal digraph homomorphisms Chaotic and periodic behavior of jeu de taquin on infinite Young tableaux Enumerating Pattern Avoiding Parking Functions Incidence toric ideals and three-point functions Unique Winning Opening Move in Three-Row Chomp Strong majority colorings of graphs A Balancing Theorem for Spanning Trees of Rectangular Grid Graphs Spectral radius and edge-disjoint connected factors of graphs New invariants for rank metric codes, with applications to the classification of rank two semifields of order 256 Flexible DP-4-coloring of planar graphs without 4-cycles and intersecting triangles Balanced intersection size distributions in projective planes List Reconstruction Problem with List Size Two Is Dimensionality a Barrier for Retrieval Models? The INIEP: Irreducible and Positive Realizations The number of Pfaffian orientations on punctured polygonally cellulated surfaces Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern Finite-state enumeration of adjacency-constrained 132-avoiding permutations
Every signed planar graph is $5$-choosable: A short proof...
Pie Desire Ebode Atangana, Maxwell Ndognkon Manga · 2026-05-20 · via math.CO updates on arXiv.org

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.