1,864 research outputs found
On small Mixed Pattern Ramsey numbers
We call the minimum order of any complete graph so that for any coloring of
the edges by colors it is impossible to avoid a monochromatic or rainbow
triangle, a Mixed Ramsey number. For any graph with edges colored from the
above set of colors, if we consider the condition of excluding in the
above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted
. We determine this function in terms of for all colored -cycles
and all colored -cliques. We also find bounds for when is a
monochromatic odd cycles, or a star for sufficiently large . We state
several open questions.Comment: 16 page
A Trichotomy for Regular Simple Path Queries on Graphs
Regular path queries (RPQs) select nodes connected by some path in a graph.
The edge labels of such a path have to form a word that matches a given regular
expression. We investigate the evaluation of RPQs with an additional constraint
that prevents multiple traversals of the same nodes. Those regular simple path
queries (RSPQs) find several applications in practice, yet they quickly become
intractable, even for basic languages such as (aa)* or a*ba*.
In this paper, we establish a comprehensive classification of regular
languages with respect to the complexity of the corresponding regular simple
path query problem. More precisely, we identify the fragment that is maximal in
the following sense: regular simple path queries can be evaluated in polynomial
time for every regular language L that belongs to this fragment and evaluation
is NP-complete for languages outside this fragment. We thus fully characterize
the frontier between tractability and intractability for RSPQs, and we refine
our results to show the following trichotomy: Evaluations of RSPQs is either
AC0, NL-complete or NP-complete in data complexity, depending on the regular
language L. The fragment identified also admits a simple characterization in
terms of regular expressions.
Finally, we also discuss the complexity of the following decision problem:
decide, given a language L, whether finding a regular simple path for L is
tractable. We consider several alternative representations of L: DFAs, NFAs or
regular expressions, and prove that this problem is NL-complete for the first
representation and PSPACE-complete for the other two. As a conclusion we extend
our results from edge-labeled graphs to vertex-labeled graphs and vertex-edge
labeled graphs.Comment: 15 pages, conference submissio
A Victorian Age Proof of the Four Color Theorem
In this paper we have investigated some old issues concerning four color map
problem. We have given a general method for constructing counter-examples to
Kempe's proof of the four color theorem and then show that all counterexamples
can be rule out by re-constructing special 2-colored two paths decomposition in
the form of a double-spiral chain of the maximal planar graph. In the second
part of the paper we have given an algorithmic proof of the four color theorem
which is based only on the coloring faces (regions) of a cubic planar maps. Our
algorithmic proof has been given in three steps. The first two steps are the
maximal mono-chromatic and then maximal dichromatic coloring of the faces in
such a way that the resulting uncolored (white) regions of the incomplete
two-colored map induce no odd-cycles so that in the (final) third step four
coloring of the map has been obtained almost trivially.Comment: 27 pages, 18 figures, revised versio
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