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arXiv · 1505.07294

Fine structure of 4-critical triangle-free graphs I. Planar graphs with two triangles and 3-colorability of chains

Zdeněk Dvořák·Bernard Lidický
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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.

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Zdeněk Dvořák, Bernard Lidický. 2015-05-27. Fine structure of 4-critical triangle-free graphs I. Planar graphs with two triangles and 3-colorability of chains. https://doi.org/10.1137/15m1023385

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