9 research outputs found
Fine structure of 4-critical triangle-free graphs I. Planar graphs with two triangles and 3-colorability of chains
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.Comment: 38 pages, 6 figures; corrections from the review proces
Embedded graph 3-coloring and flows
A graph drawn in a surface is a near-quadrangulation if the sum of the
lengths of the faces different from 4-faces is bounded by a fixed constant. We
leverage duality between colorings and flows to design an efficient algorithm
for 3-precoloring-extension in near-quadrangulations of orientable surfaces.
Furthermore, we use this duality to strengthen previously known sufficient
conditions for 3-colorability of triangle-free graphs drawn in orientable
surfaces.Comment: 53 pages, 15 figure