11 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
-Critical Graphs in -Free Graphs
Given two graphs and , a graph is -free if it
contains no induced subgraph isomorphic to or . Let be the
path on vertices. A graph is -vertex-critical if has chromatic
number but every proper induced subgraph of has chromatic number less
than . The study of -vertex-critical graphs for graph classes is an
important topic in algorithmic graph theory because if the number of such
graphs that are in a given hereditary graph class is finite, then there is a
polynomial-time algorithm to decide if a graph in the class is
-colorable.
In this paper, we initiate a systematic study of the finiteness of
-vertex-critical graphs in subclasses of -free graphs. Our main result
is a complete classification of the finiteness of -vertex-critical graphs in
the class of -free graphs for all graphs on 4 vertices. To obtain
the complete dichotomy, we prove the finiteness for four new graphs using
various techniques -- such as Ramsey-type arguments and the dual of Dilworth's
Theorem -- that may be of independent interest.Comment: 18 page
Obstructions for three-coloring graphs without induced paths on six vertices
We prove that there are 24 4-critical P-6-free graphs, and give the complete list. We remark that, if H is connected and not a subgraph of P-6, there are infinitely many 4-critical H-free graphs. Our result answers questions of Golovach et al. and Seymour. (C) 2019 Published by Elsevier Inc
Colouring graphs with no induced six-vertex path or diamond
The diamond is the graph obtained by removing an edge from the complete graph
on 4 vertices. A graph is (, diamond)-free if it contains no induced
subgraph isomorphic to a six-vertex path or a diamond. In this paper we show
that the chromatic number of a (, diamond)-free graph is no larger
than the maximum of 6 and the clique number of . We do this by reducing the
problem to imperfect (, diamond)-free graphs via the Strong Perfect Graph
Theorem, dividing the imperfect graphs into several cases, and giving a proper
colouring for each case. We also show that there is exactly one
6-vertex-critical (, diamond, )-free graph. Together with the
Lov\'asz theta function, this gives a polynomial time algorithm to compute the
chromatic number of (, diamond)-free graphs.Comment: 29 page