121 research outputs found
Subdivisions of a large clique in C6-free graphs
Mader conjectured that every -free graph has a subdivision of a clique of order linear in its average degree. We show that every -free graph has such a subdivision of a large clique.
We also prove the dense case of Mader's conjecture in a stronger sense, i.e., for every c, there is a such that every -free graph with average degree has a subdivision of a clique with where every edge is subdivided exactly 3 times
A proof of Mader's conjecture on large clique subdivisions in -free graphs
Given any integers , we show there exists some such
that any -free graph with average degree contains a subdivision of
a clique with at least vertices. In particular,
when this resolves in a strong sense the conjecture of Mader in 1999 that
every -free graph has a subdivision of a clique with order linear in the
average degree of the original graph. In general, the widely conjectured
asymptotic behaviour of the extremal density of -free graphs suggests
our result is tight up to the constant .Comment: 25 pages, 1 figur
Clique complexes and graph powers
We study the behaviour of clique complexes of graphs under the operation of
taking graph powers. As an example we compute the clique complexes of powers of
cycles, or, in other words, the independence complexes of circular complete
graphs.Comment: V3: final versio
Maximum Independent Sets in Subcubic Graphs: New Results
The maximum independent set problem is known to be NP-hard in the class of
subcubic graphs, i.e. graphs of vertex degree at most 3. We present a
polynomial-time solution in a subclass of subcubic graphs generalizing several
previously known results
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
Rainbow clique subdivisions
We show that for any integer , every properly edge colored
-vertex graph with average degree at least contains a
rainbow subdivision of a complete graph of size . Note that this bound is
within a log factor of the lower bound. This also implies a result on the
rainbow Tur\'{a}n number of cycles
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