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Planar graphs with distance of 3-cycles at least 2 and no cycles of lengths 5, 6, 7

Tao Wang, Ya-Nan Wang, Xiaojing Yang

TL;DR

This work addresses planar graphs with $\mathrm{dist}^{\Delta} \geq 2$ and no $5$-, $6$-, or $7$-cycles, proving they are weakly $2$-degenerate and $(\mathcal{I},\mathcal{F})$-partitionable. It develops a structural lemma (STR) via a discharging process to rule out minimal counterexamples, and then leverages the Bernshteyn–Lee weak-degeneracy framework with valued covers to show no counterexample exists to $\mathrm{wd}(G)\le 2$. The final step connects these ideas to strictly $f$-degenerate transversals (SfDT), showing that every such graph admits a partition into an independent set and a forest. Overall, the results yield new sufficient conditions for weak degeneracy in planar graphs and provide a constructive $(\mathcal{I},\mathcal{F})$-partition, with implications for related coloring parameters via WD bounds.

Abstract

Weak degeneracy of a graph is a variation of degeneracy that has a close relationship to many graph coloring parameters. In this article, we prove that planar graphs with distance of $3$-cycles at least 2 and no cycles of lengths $5, 6, 7$ are weakly $2$-degenerate. Furthermore, such graphs can be vertex-partitioned into two subgraphs, one of which has no edges, and the other is a forest.

Planar graphs with distance of 3-cycles at least 2 and no cycles of lengths 5, 6, 7

TL;DR

This work addresses planar graphs with and no -, -, or -cycles, proving they are weakly -degenerate and -partitionable. It develops a structural lemma (STR) via a discharging process to rule out minimal counterexamples, and then leverages the Bernshteyn–Lee weak-degeneracy framework with valued covers to show no counterexample exists to . The final step connects these ideas to strictly -degenerate transversals (SfDT), showing that every such graph admits a partition into an independent set and a forest. Overall, the results yield new sufficient conditions for weak degeneracy in planar graphs and provide a constructive -partition, with implications for related coloring parameters via WD bounds.

Abstract

Weak degeneracy of a graph is a variation of degeneracy that has a close relationship to many graph coloring parameters. In this article, we prove that planar graphs with distance of -cycles at least 2 and no cycles of lengths are weakly -degenerate. Furthermore, such graphs can be vertex-partitioned into two subgraphs, one of which has no edges, and the other is a forest.

Paper Structure

This paper contains 4 sections, 11 theorems, 3 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proof of \ref{['STR']}
  3. Proof of \ref{['Weak']}
  4. Proof of \ref{['IF:partition']}

Key Result

Theorem 1.1

If $G$ is a triangle-free planar graph in which no 4-cycle is normally adjacent to a cycle of length at most five, then $\textsf{wd}(G) \leq 2$.

Figures (4)

  • Figure 1: A set of unavoidable configurations, where a semicircle represents a path of length two or three with interior vertices having degree three in $G$, a solid point represents a vertex of degree three, and a green square point represents a vertex of degree four in $G$.
  • Figure 2: A $7$-face adjacent to a $4$-face.
  • Figure 3: Two types of special $8$-faces.
  • Figure 5: A set of unavoidable configurations, where a semicircle represents a path of length two or three with interior vertices having degree three in $G$, a solid point represents a vertex of degree three, and a green square point represents a vertex of degree four in $G$.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Theorem 1.1: Han et al. MR4663366
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 3.1
  • ...and 3 more