---
abstract: 'Aksenov proved that in a planar graph $G$ with at most one triangle, every precoloring of a $4$-cycle can be extended to a $3$-coloring of $G$. We give an exact characterization of planar graphs with two triangles in that some precoloring of a $4$-cycle does not extend. We apply this characterization to solve the precoloring extension problem from two $4$-cycles in a triangle-free planar graph in the case that the precolored $4$-cycles are separated by many disjoint $4$-cycles. The latter result is used in followup papers to give detailed information about the structure of $4$-critical triangle-free graphs embedded in a fixed surface.'
author:
- 'Zdeněk Dvořák[^1]'
- 'Bernard Lidický[^2]'
bibliography:
- 'cylgen.bib'
title: 'Fine structure of $4$-critical triangle-free graphs I. Planar graphs with two triangles and $3$-colorability of chains'
---

Introduction
============

The interest in the $3$-coloring properties of planar graphs was started by a celebrated theorem of Grötzsch [@grotzsch1959], who proved that every planar triangle-free graph is $3$-colorable. While in general, deciding $3$-colorablity of a planar graph is a NP-complete problem [@garey1979computers], there are many other sufficient conditions guaranteeing $3$-colorability, see e.g. the survey of Montassier [@montasweb].

For a long time, the question of the complexity of 