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2-distance 4-coloring of planar subcubic graphs with girth at least 21

Hoang La, Mickael Montassier

TL;DR

The paper addresses the problem of $2$-distance coloring in planar graphs with maximum degree $3$ by focusing on girth constraints. It proves that every planar subcubic graph with girth at least $21$ admits a $2$-distance $4$-coloring, using a minimal counterexample and a detailed discharging method to exclude all possible obstructions. In addition, it provides a construction of a planar subcubic graph with girth $11$ that is not $2$-distance $4$-colorable, establishing a lower bound on the girth required for the $4$-coloring guarantee. Together, these results refine the understanding of Wegner-type bounds for sparse planar graphs and illustrate the role of high girth in enabling tight $2$-distance colorings; the gadget-based constructions underpin both the positive result and the lower-bound example.

Abstract

A $2$-distance $k$-coloring of a graph is a proper vertex $k$-coloring where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance $4$-coloring for planar subcubic graphs with girth at least 21. We also show a construction of a planar subcubic graph of girth 11 that is not $2$-distance $4$-colorable.

2-distance 4-coloring of planar subcubic graphs with girth at least 21

TL;DR

The paper addresses the problem of -distance coloring in planar graphs with maximum degree by focusing on girth constraints. It proves that every planar subcubic graph with girth at least admits a -distance -coloring, using a minimal counterexample and a detailed discharging method to exclude all possible obstructions. In addition, it provides a construction of a planar subcubic graph with girth that is not -distance -colorable, establishing a lower bound on the girth required for the -coloring guarantee. Together, these results refine the understanding of Wegner-type bounds for sparse planar graphs and illustrate the role of high girth in enabling tight -distance colorings; the gadget-based constructions underpin both the positive result and the lower-bound example.

Abstract

A -distance -coloring of a graph is a proper vertex -coloring where vertices at distance at most 2 cannot share the same color. We prove the existence of a -distance -coloring for planar subcubic graphs with girth at least 21. We also show a construction of a planar subcubic graph of girth 11 that is not -distance -colorable.

Paper Structure

This paper contains 8 sections, 16 theorems, 4 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorem 4
  3. Notations and drawing conventions.
  4. Useful observations
  5. Structural properties of G
  6. Discharging rules
  7. Verifying that charges on each face and each vertex are non-negative
  8. A non 4-colorable subcubic planar graph of girth 11

Key Result

Proposition 2

For every planar graph $G$, $(\mathop{\mathrm{mad}}\nolimits(G)-2)(g(G)-2)<4$.

Figures (15)

  • Figure 1: Examples of Moore graphs for which $\chi^2=\Delta^2+1$.
  • Figure 2: Graphs with $\chi^2\approx \frac{3}{2} \Delta$
  • Figure 3: An useful non-colorable graph on three vertices.
  • Figure 4: Graph $H$ from \ref{['restriction lemma']}.
  • Figure 5: Colorable graphs.
  • ...and 10 more figures

Theorems & Definitions (30)

  • Conjecture 1: wegner
  • Proposition 2: Folklore
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 20 more