8,673 research outputs found
Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
Chromatic thresholds in dense random graphs
The chromatic threshold of a graph with respect to the
random graph is the infimum over such that the following holds
with high probability: the family of -free graphs with
minimum degree has bounded chromatic number. The study of
the parameter was initiated in 1973 by
Erd\H{o}s and Simonovits, and was recently determined for all graphs . In
this paper we show that for all fixed , but that typically if . We also make significant progress towards determining
for all graphs in the range . In sparser random graphs the
problem is somewhat more complicated, and is studied in a separate paper.Comment: 36 pages (including appendix), 1 figure; the appendix is copied with
minor modifications from arXiv:1108.1746 for a self-contained proof of a
technical lemma; accepted to Random Structures and Algorithm
Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm
Many practical problems in almost all scientific and technological
disciplines have been classified as computationally hard (NP-hard or even
NP-complete). In life sciences, combinatorial optimization problems frequently
arise in molecular biology, e.g., genome sequencing; global alignment of
multiple genomes; identifying siblings or discovery of dysregulated pathways.In
almost all of these problems, there is the need for proving a hypothesis about
certain property of an object that can be present only when it adopts some
particular admissible structure (an NP-certificate) or be absent (no admissible
structure), however, none of the standard approaches can discard the hypothesis
when no solution can be found, since none can provide a proof that there is no
admissible structure. This article presents an algorithm that introduces a
novel type of solution method to "efficiently" solve the graph 3-coloring
problem; an NP-complete problem. The proposed method provides certificates
(proofs) in both cases: present or absent, so it is possible to accept or
reject the hypothesis on the basis of a rigorous proof. It provides exact
solutions and is polynomial-time (i.e., efficient) however parametric. The only
requirement is sufficient computational power, which is controlled by the
parameter . Nevertheless, here it is proved that the
probability of requiring a value of to obtain a solution for a
random graph decreases exponentially: , making
tractable almost all problem instances. Thorough experimental analyses were
performed. The algorithm was tested on random graphs, planar graphs and
4-regular planar graphs. The obtained experimental results are in accordance
with the theoretical expected results.Comment: Working pape
Bipartite induced density in triangle-free graphs
We prove that any triangle-free graph on vertices with minimum degree at
least contains a bipartite induced subgraph of minimum degree at least
. This is sharp up to a logarithmic factor in . Relatedly, we show
that the fractional chromatic number of any such triangle-free graph is at most
the minimum of and as . This is
sharp up to constant factors. Similarly, we show that the list chromatic number
of any such triangle-free graph is at most as
.
Relatedly, we also make two conjectures. First, any triangle-free graph on
vertices has fractional chromatic number at most
as . Second, any triangle-free
graph on vertices has list chromatic number at most as
.Comment: 20 pages; in v2 added note of concurrent work and one reference; in
v3 added more notes of ensuing work and a result towards one of the
conjectures (for list colouring
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