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The square of a subcubic planar graph without a 5-cycle is 7-choosable

Seog-Jin Kim, Xiaopan Lian, Atsuhiro Nakamoto, Kenta Ozeki

TL;DR

The paper proves that the square of any subcubic planar graph with no 5-cycles is 7-choosable, advancing beyond prior results that required stronger constraints on girth or forbidden cycles. It combines a discharging framework with Combinatorial Nullstellensatz-based coloring arguments to exclude a family of reducible configurations in any minimal counterexample. The key contribution is showing $\chi_{\ell}(G^2) \le 7$ under the single constraint of forbidding 5-cycles, thereby extending previous results and informing Wegner-type questions for list colorings of graph squares. This result has potential implications for understanding colorability properties of planar graphs and their squares in broader combinatorial settings.

Abstract

The square of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Thomassen [12] showed that $χ(G^2) \leq 7$ if $G$ is a subcubic planar graph. A natural question is whether $χ_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph. Recently Kim and Lian [11] showed that $χ_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. And Jin, Kang, and Kim [10] showed that $χ_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph without 4-cycles and 5-cycles. In this paper, we show that the square of a subcubic planar graph without 5-cycles is 7-choosable, which improves the results of [10] and [11].

The square of a subcubic planar graph without a 5-cycle is 7-choosable

TL;DR

The paper proves that the square of any subcubic planar graph with no 5-cycles is 7-choosable, advancing beyond prior results that required stronger constraints on girth or forbidden cycles. It combines a discharging framework with Combinatorial Nullstellensatz-based coloring arguments to exclude a family of reducible configurations in any minimal counterexample. The key contribution is showing under the single constraint of forbidding 5-cycles, thereby extending previous results and informing Wegner-type questions for list colorings of graph squares. This result has potential implications for understanding colorability properties of planar graphs and their squares in broader combinatorial settings.

Abstract

The square of a graph , denoted , has the same vertex set as and has an edge between two vertices if the distance between them in is at most . Thomassen [12] showed that if is a subcubic planar graph. A natural question is whether or not if is a subcubic planar graph. Recently Kim and Lian [11] showed that if is a subcubic planar graph of girth at least 6. And Jin, Kang, and Kim [10] showed that if is a subcubic planar graph without 4-cycles and 5-cycles. In this paper, we show that the square of a subcubic planar graph without 5-cycles is 7-choosable, which improves the results of [10] and [11].
Paper Structure (11 sections, 23 theorems, 67 equations, 35 figures)

This paper contains 11 sections, 23 theorems, 67 equations, 35 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Summary of reducible configurations
  4. Proof of Theorem \ref{['main-thm']}
  5. Proofs of reducible configurations
  6. Basic configurations
  7. Special configurations
  8. Subgraph $H_2$ is reducible
  9. Subgraph $H_3$ is reducible
  10. Subgraph $H_4$ is reducible
  11. 7-face is adjacent to at most one 4-face

Key Result

Theorem 3

If $G$ is a subcubic planar graph without 5-cycles, then $\chi_{\ell}(G^2) \leq 7$.

Figures (35)

  • Figure 4: Discharging rules
  • Figure 5: Graph $T$
  • Figure 6: Cases 1 and 2 of $H_1$, respectively. The numbers at vertices are the number of available colors.
  • Figure 7: A 6-cycle $C$ which has a $2$-vertex $v_6$.
  • Figure 8: The distance between $3$-face and $4$-face is 1. The numbers at vertices are the number of available colors.
  • ...and 30 more figures

Theorems & Definitions (45)

  • Conjecture 1
  • Theorem 3
  • Corollary 5
  • Theorem 6: Alon1999
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • Lemma 9
  • proof
  • ...and 35 more