21 research outputs found
3-coloring triangle-free planar graphs with a precolored 8-cycle
Let G be a planar triangle-free graph and let C be a cycle in G of length at
most 8. We characterize all situations where a 3-coloring of C does not extend
to a proper 3-coloring of the whole graph.Comment: 20 pages, 5 figure
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