The Alon-Tarsi Number of Squares of Subcubic Planar Graphs without Cycles of Lengths $4$ to $8$

Abstract: The Alon--Tarsi number $AT(G)$ of a graph $G$, defined via the graph polynomial, is a strengthening of the list chromatic number $χ<em>{\ell}(G)$. We study the Alon--Tarsi number of squares of planar graphs. The square of a graph $G$ is the graph obtained by joining every pair of vertices whose distance in $G$ is at most $2$. Recently, Kim and Luo (2026) proved that $χ</em>{\ell}(G^2)\le 6$ for every subcubic planar graph containing no $k$-cycles for $4\le k\le 8$. We strengthen this result by proving that $AT(G^2)\le 6$ for every such graph $G$.

• The paper establishes that the square of any subcubic planar graph without cycles of lengths 4–8 has an Alon–Tarsi number of at most 6.
• It employs a blend of polynomial methods, orientation techniques, and discharging arguments to verify the main result.
• The result provides an explicit algebraic certificate for list coloring with implications for distributed algorithms and further theoretical research.

The Alon–Tarsi Number of Squares of Subcubic Planar Graphs without Cycles of Lengths $4$ to $8$

Introduction and Context

This paper addresses a refinement of combinatorial coloring parameters in planar graphs, focusing on the Alon–Tarsi number ($AT(G)$), an important strengthening of the list chromatic number ($\chi_\ell(G)$) via the Alon–Tarsi polynomial method and orientation techniques. The specific graph class considered consists of subcubic planar graphs (planar graphs of maximum degree 3) that omit cycles of lengths 4 through 8. The main object of study is the square of such graphs, $G^2$, in which two vertices are adjacent if their distance in $G$ is at most 2.

Wegner's conjecture and subsequent work have established the difficulty of sharply bounding the chromatic and list chromatic numbers of $G^2$ in various classes of planar graphs. Recent work by Kim and Luo provided the bound $\chi_\ell(G^2) \leq 6$ for the class of graphs of interest. The contribution of this paper is to demonstrate that the Alon–Tarsi number of the square also adheres to the same upper bound, i.e., $AT(G^2) \leq 6$, and thus that this bound witnesses an explicit algebraic coloring certificate.

Alon–Tarsi Number, Orientations, and Polynomial Methodologies

The Alon–Tarsi number generalizes list coloring by utilizing the nonvanishing of certain coefficients in the graph polynomial

$P_G(\bm{x})=\prod_{u\sim v,\;u<v}(x_u-x_v)$

under vertex orderings. This connects to feasible colorings over arbitrary list assignments (choosability). The logical framework of the Alon–Tarsi method employs the Combinatorial Nullstellensatz, reducing coloring to the existence of certain monomials with nonzero coefficients. This yields $8$0, and an upper bound on $8$1 is strictly stronger than a corresponding bound on $8$2.

Moreover, the Alon–Tarsi number can be equivalently characterized in orientation-theoretic terms: a graph is Alon–Tarsi $8$3-choosable if it admits an orientation whose out-degree at each vertex is at most $8$4 and for which the signed difference in Eulerian subdigraphs (even minus odd) is nonzero. The connection between the algebraic and orientation frameworks enables powerful induction and reducibility arguments.

Main Theorem and Structural Approach

Main result: For any subcubic planar graph $8$5 with no $8$6-cycles for $8$7, $8$8.

The proof strategy mirrors approaches from the study of list coloring in planar graphs but introduces new reducibility arguments suitable for the orientation method. The proof is by contradiction, assuming a minimal counterexample, and proceeds in several stages:

1. Oriented Minimal Obstruction: If a minimal $8$9 exists with $AT(G)$0, certain properties are induced: minimum degree at least two, no two adjacent degree-two vertices, exclusion of certain configurations containing triangles, and so forth.
2. Structural Reducibility: The core of the argument is to establish that subgraphs (corresponding to faces or possible induced subgraphs that might violate $AT(G)$1) are in fact reducible, i.e., they can be shown to be Alon–Tarsi choosable under specific boundary color list sizes determined by their "cuts" with the rest of $AT(G)$2. This is achieved by careful computation of relevant coefficients in the graph polynomial (often aided by computer algebra systems) and application of the orientation characterizations.
3. Discharging Argument: With the reducibility analysis establishing that all "dangerous" configurations are avoidable, a discharging method is launched, assigning initial charges to vertices and faces and redistributing them via standard planar discharging rules. The potential for a negative total charge unless the claimed upper bound is respected implies the contradiction needed to eliminate the minimal counterexample.

The argument extends bounds on configurations involving cycles and faces by careful enumeration and systematic elimination of each configuration via explicit algebraic verification.

Strong Numerical Results and Contradictory Claims

The main numerical result is the assertion that if $AT(G)$3 is subcubic, planar, and contains no cycles of length 4 through 8, then the square of $AT(G)$4 satisfies $AT(G)$5. This strictly strengthens the list coloring result in [KLuo25] by certifying a polynomially checkable coloring via the Alon–Tarsi method. The paper does not assert that the bound of 6 is best possible but shows through discharging and reducibility that no counterexample with $AT(G)$6 exists in the specified class.

Practical and Theoretical Implications

On the practical side, an upper bound on $AT(G)$7 gives an explicit witness for polynomial list coloring algorithms derived from algebraic tools, which has implications for distributed algorithms and constructive proofs of list colorability. It provides a roadmap for handling list coloring and its refinements in similar minor-closed classes of graphs where structural obstructions are already well understood.

Theoretically, this work demonstrates the robustness of algebraic coloring methods and their ability to subsume combinatorial results via more powerful invariants. The careful combinatorial and algebraic analysis of low-degree, cycle-free planar graphs consolidates the link between the orientation-based approaches and classical coloring. It also suggests further gains may potentially be obtained for higher-degree classes or for planar graphs with milder girth restrictions, contingent on more sophisticated reducibility and discharging arguments.

Further research could target improving the bound to the true value of $AT(G)$8 in this (or related) classes, or relaxing the cycle-length restrictions. Additionally, extending these techniques to other classes of minor-closed families or hybrid planar constructions may reveal deeper algebraic-combinatorial correspondences.

Conclusion

This paper provides a significant technical advance in the study of refined graph coloring parameters, showing that the Alon–Tarsi number of squares of subcubic planar graphs without small cycles is bounded above by 6. The methodology—relying on the interplay between orientation characterizations, the Combinatorial Nullstellensatz, and explicit reducibility/discharging—furnishes a strong platform for future work on algebraic coloring bounds in planar and related graph families, enriching the toolkit for both theoretical and algorithmic coloring research (2607.04494).