287 research outputs found
Exhaustive generation of -critical -free graphs
We describe an algorithm for generating all -critical -free
graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove
that there are only finitely many -critical -free graphs, for
both and . We also show that there are only finitely many
-critical graphs -free graphs. For each case of these cases we
also give the complete lists of critical graphs and vertex-critical graphs.
These results generalize previous work by Hell and Huang, and yield certifying
algorithms for the -colorability problem in the respective classes.
Moreover, we prove that for every , the class of 4-critical planar
-free graphs is finite. We also determine all 27 4-critical planar
-free graphs.
We also prove that every -free graph of girth at least five is
3-colorable, and determine the smallest 4-chromatic -free graph of
girth five. Moreover, we show that every -free graph of girth at least
six and every -free graph of girth at least seven is 3-colorable. This
strengthens results of Golovach et al.Comment: 17 pages, improved girth results. arXiv admin note: text overlap with
arXiv:1504.0697
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
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