2,260 research outputs found
List precoloring extension in planar graphs
A celebrated result of Thomassen states that not only can every planar graph
be colored properly with five colors, but no matter how arbitrary palettes of
five colors are assigned to vertices, one can choose a color from the
corresponding palette for each vertex so that the resulting coloring is proper.
This result is referred to as 5-choosability of planar graphs. Albertson asked
whether Thomassen's theorem can be extended by precoloring some vertices which
are at a large enough distance apart in a graph. Here, among others, we answer
the question in the case when the graph does not contain short cycles
separating precolored vertices and when there is a "wide" Steiner tree
containing all the precolored vertices.Comment: v2: 15 pages, 11 figres, corrected typos and new proof of Theorem
3(2
Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count
We show that triangle-free penny graphs have degeneracy at most two, list
coloring number (choosability) at most three, diameter , and
at most edges.Comment: 10 pages, 2 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Fractional coloring of triangle-free planar graphs
We prove that every planar triangle-free graph on vertices has fractional
chromatic number at most
- …