24 research outputs found
5-list-coloring planar graphs with distant precolored vertices
We answer positively the question of Albertson asking whether every planar
graph can be -list-colored even if it contains precolored vertices, as long
as they are sufficiently far apart from each other. In order to prove this
claim, we also give bounds on the sizes of graphs critical with respect to
5-list coloring. In particular, if G is a planar graph, H is a connected
subgraph of G and L is an assignment of lists of colors to the vertices of G
such that |L(v)| >= 5 for every v in V(G)-V(H) and G is not L-colorable, then G
contains a subgraph with O(|H|^2) vertices that is not L-colorable.Comment: 53 pages, 9 figures version 2: addresses suggestions by reviewer
Facial unique-maximum colorings of plane graphs with restriction on big vertices
A facial unique-maximum coloring of a plane graph is a proper coloring of the
vertices using positive integers such that each face has a unique vertex that
receives the maximum color in that face. Fabrici and G\"{o}ring (2016) proposed
a strengthening of the Four Color Theorem conjecturing that all plane graphs
have a facial unique-maximum coloring using four colors. This conjecture has
been disproven for general plane graphs and it was shown that five colors
suffice. In this paper we show that plane graphs, where vertices of degree at
least four induce a star forest, are facially unique-maximum 4-colorable. This
improves a previous result for subcubic plane graphs by Andova, Lidick\'y,
Lu\v{z}ar, and \v{S}krekovski (2018). We conclude the paper by proposing some
problems.Comment: 8 pages, 5 figure
Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies
We settle a problem of Havel by showing that there exists an absolute
constant d such that if G is a planar graph in which every two distinct
triangles are at distance at least d, then G is 3-colorable. In fact, we prove
a more general theorem. Let G be a planar graph, and let H be a set of
connected subgraphs of G, each of bounded size, such that every two distinct
members of H are at least a specified distance apart and all triangles of G are
contained in \bigcup{H}. We give a sufficient condition for the existence of a
3-coloring phi of G such that for every B\in H, the restriction of phi to B is
constrained in a specified way.Comment: 26 pages, no figures. Updated presentatio
A general framework for coloring problems: old results, new results, and open problems
In this survey paper we present a general framework for coloring problems that was introduced in a joint paper which the author presented at WG2003. We show how a number of different types of coloring problems, most of which have been motivated from frequency assignment, fit into this framework. We give a survey of the existing results, mainly based on and strongly biased by joint work of the author with several different groups of coauthors, include some new results, and discuss several open problems for each of the variants
5-choosability of graphs with crossings far apart
We give a new proof of the fact that every planar graph is 5-choosable, and
use it to show that every graph drawn in the plane so that the distance between
every pair of crossings is at least 15 is 5-choosable. At the same time we may
allow some vertices to have lists of size four only, as long as they are far
apart and far from the crossings.Comment: 55 pages, 11 figures; minor revision according to the referee
suggestion
Fractional coloring of triangle-free planar graphs
We prove that every planar triangle-free graph on vertices has fractional
chromatic number at most