17 research outputs found
b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs
A b-coloring of a graph is a proper coloring such that every color class
contains a vertex that is adjacent to all other color classes. The b-chromatic
number of a graph G, denoted by \chi_b(G), is the maximum number t such that G
admits a b-coloring with t colors. A graph G is called b-continuous if it
admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and
b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of
G, and every induced subgraph H_2 of H_1.
We investigate the b-chromatic number of graphs with stability number two.
These are exactly the complements of triangle-free graphs, thus including all
complements of bipartite graphs. The main results of this work are the
following:
- We characterize the b-colorings of a graph with stability number two in
terms of matchings with no augmenting paths of length one or three. We derive
that graphs with stability number two are b-continuous and b-monotonic.
- We prove that it is NP-complete to decide whether the b-chromatic number of
co-bipartite graph is at most a given threshold.
- We describe a polynomial time dynamic programming algorithm to compute the
b-chromatic number of co-trees.
- Extending several previous results, we show that there is a polynomial time
dynamic programming algorithm for computing the b-chromatic number of
tree-cographs. Moreover, we show that tree-cographs are b-continuous and
b-monotonic
Approximately coloring graphs without long induced paths
It is an open problem whether the 3-coloring problem can be solved in
polynomial time in the class of graphs that do not contain an induced path on
vertices, for fixed . We propose an algorithm that, given a 3-colorable
graph without an induced path on vertices, computes a coloring with
many colors. If the input graph is
triangle-free, we only need many
colors. The running time of our algorithm is if the input
graph has vertices and edges
-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
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
Clique-width : harnessing the power of atoms.
Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class G if they are so on the atoms (graphs with no clique cut-set) of G . Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph G is H-free if H is not an induced subgraph of G, and it is (H1,H2) -free if it is both H1 -free and H2 -free. A class of H-free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for (H1,H2) -free graphs, as evidenced by one known example. We prove the existence of another such pair (H1,H2) and classify the boundedness of clique-width on (H1,H2) -free atoms for all but 18 cases
k-Critical graphs in P5-free graphs
Given two graphs H1 and H2, a graph G is (H1, H2)-free if it contains no induced subgraph isomorphic to H1 or H2. Let Pt be the path on t vertices. A graph G is k-vertex-critical if G has chromatic number k but every proper induced subgraph of G has chromatic number less than k. The study of k-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 (k−1)- colorable. In this paper, we initiate a systematic study of the finiteness of k-vertex-critical graphs in subclasses of P5-free graphs. Our main result is a complete classification of the finiteness of k-vertex-critical graphs in the class of (P5, H)-free graphs for all graphs H on 4 vertices. To obtain the complete dichotomy, we prove the finiteness for four new graphs H using various techniques – such as Ramsey-type arguments and the dual of Dilworth’s Theorem – that may be of independent interest
Approximately Coloring Graphs Without Long Induced Paths
It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on t vertices, for fixed t. We propose an algorithm that, given a 3-colorable graph without an induced path on t vertices, computes a coloring with max{5,2⌈t-12⌉-2} many colors. If the input graph is triangle-free, we only need max{4,⌈t-12⌉+1} many colors. The running time of our algorithm is O((3 t - 2+ t2) m+ n) if the input graph has n vertices and m edges