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11/4-colorability of subcubic triangle-free graphs

Zdeněk Dvořák, Bernard Lidický, Luke Postle

TL;DR

This work establishes that every connected subcubic triangle-free graph has fractional chromatic number at most $\tfrac{11}{4}$, with exactly two exceptional graphs where the bound is tight. The authors develop a robust LP-based framework using e-graphs, define $11/4$-colorings, and implement a computer-assisted enumeration of a finite set of critical graphs to support a constructive inductive coloring argument. A central strategy is to rule out all potential minimal counterexamples by progressively eliminating structural configurations (two-degree vertices, $K_4^+$, 4-cycles, etc.), culminating in a 3-regular conclusion and a global convex-combination coloring scheme. The result implies the same bound for subcubic triangle-free planar graphs and advances the broader conjecture linking planarity, triangle-freeness, and fractional colorability, providing a concrete pathway toward tighter planar bounds via reducibility and LP methods.

Abstract

We prove that up to two exceptions, every connected subcubic triangle-free graph has fractional chromatic number at most 11/4. This is tight unless further exceptional graphs are excluded, and improves the known bound on the fractional chromatic number of subcubic triangle-free planar graphs.

11/4-colorability of subcubic triangle-free graphs

TL;DR

This work establishes that every connected subcubic triangle-free graph has fractional chromatic number at most , with exactly two exceptional graphs where the bound is tight. The authors develop a robust LP-based framework using e-graphs, define -colorings, and implement a computer-assisted enumeration of a finite set of critical graphs to support a constructive inductive coloring argument. A central strategy is to rule out all potential minimal counterexamples by progressively eliminating structural configurations (two-degree vertices, , 4-cycles, etc.), culminating in a 3-regular conclusion and a global convex-combination coloring scheme. The result implies the same bound for subcubic triangle-free planar graphs and advances the broader conjecture linking planarity, triangle-freeness, and fractional colorability, providing a concrete pathway toward tighter planar bounds via reducibility and LP methods.

Abstract

We prove that up to two exceptions, every connected subcubic triangle-free graph has fractional chromatic number at most 11/4. This is tight unless further exceptional graphs are excluded, and improves the known bound on the fractional chromatic number of subcubic triangle-free planar graphs.
Paper Structure (12 sections, 35 theorems, 20 equations, 43 figures)

This paper contains 12 sections, 35 theorems, 20 equations, 43 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coloring and reducibility
  4. Common arguments
  5. Connectivity and nailed vertices
  6. Two neighbors of degree two
  7. Subdivided $K_4$
  8. Adjacent vertices of degree two
  9. 4-cycle with adjacent 2-vertex
  10. Vertices of degree two
  11. 4-cycles
  12. 3-regular graphs

Key Result

Theorem 1.4

Let $G$ be a subcubic triangle-free graph. If no component of $G$ is isomorphic to the graphs $F_{14}^{(1)}$ and $F_{14}^{(2)}$ depicted in Figure fig-forb, then $G$ has fractional chromatic number at most $11/4$.

Figures (43)

  • Figure 1: The graph $K_4^+$.
  • Figure 2: Valid critical cubic e-graphs $F_{14}^{(1)}$ and $F_{14}^{(2)}$ .
  • Figure 3: Subcubic graphs $F_{22}$ and $F_{11}$ with fractional chromatic number $\frac{11}{4}$ and $F_{19}^{(1)}$ and $F_{19}^{(2)}$ with fractional chromatic number $\frac{19}{7}$.
  • Figure 4: Replacing a sub-e-graph matching $F$ by $R$.
  • Figure 5: The reducible configuration $G_3$ and the replacement graph $H_3$.
  • ...and 38 more figures

Theorems & Definitions (81)

  • Conjecture 1.1: Heckman and Thomas thoheck
  • Conjecture 1.2: Heckman and Thomas HeTh06
  • Conjecture 1.3: Cames van Batenburg et al. van2019large
  • Theorem 1.4
  • Corollary 1.5
  • Conjecture 1.6
  • Definition 2.1
  • Theorem 2.3
  • proof : Proof of Theorem \ref{['thm-mainfr']}.
  • Lemma 2.4
  • ...and 71 more