1,935 research outputs found
Fine structure of 4-critical triangle-free graphs I. Planar graphs with two triangles and 3-colorability of chains
Aksenov proved that in a planar graph G with at most one triangle, every
precoloring of a 4-cycle can be extended to a 3-coloring of G. We give an exact
characterization of planar graphs with two triangles in that some precoloring
of a 4-cycle does not extend. We apply this characterization to solve the
precoloring extension problem from two 4-cycles in a triangle-free planar graph
in the case that the precolored 4-cycles are separated by many disjoint
4-cycles. The latter result is used in followup papers to give detailed
information about the structure of 4-critical triangle-free graphs embedded in
a fixed surface.Comment: 38 pages, 6 figures; corrections from the review proces
Vertex elimination orderings for hereditary graph classes
We provide a general method to prove the existence and compute efficiently
elimination orderings in graphs. Our method relies on several tools that were
known before, but that were not put together so far: the algorithm LexBFS due
to Rose, Tarjan and Lueker, one of its properties discovered by Berry and
Bordat, and a local decomposition property of graphs discovered by Maffray,
Trotignon and Vu\vskovi\'c. We use this method to prove the existence of
elimination orderings in several classes of graphs, and to compute them in
linear time. Some of the classes have already been studied, namely
even-hole-free graphs, square-theta-free Berge graphs, universally signable
graphs and wheel-free graphs. Some other classes are new. It turns out that all
the classes that we study in this paper can be defined by excluding some of the
so-called Truemper configurations. For several classes of graphs, we obtain
directly bounds on the chromatic number, or fast algorithms for the maximum
clique problem or the coloring problem
On hereditary graph classes defined by forbidding Truemper configurations: recognition and combinatorial optimization algorithms, and χ-boundedness results
Truemper configurations are four types of graphs that helped us understand the structure of several well-known hereditary graph classes. The most famous examples are perhaps the class of perfect graphs and the class of even-hole-free graphs: for both of them, some Truemper configurations are excluded (as induced subgraphs), and this fact appeared to be useful, and played some role in the proof of the known decomposition theorems for these classes.
The main goal of this thesis is to contribute to the systematic exploration of hereditary graph classes defined by forbidding Truemper configurations. We study many of these classes, and we investigate their structure by applying the decomposition method. We then use our structural results to analyze the complexity of the maximum clique, maximum stable set and optimal coloring problems restricted to these classes. Finally, we provide polynomial-time recognition algorithms for all of these classes, and we obtain χ-boundedness results
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