30 research outputs found
Cuts in matchings of 3-connected cubic graphs
We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette,
Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and
on even graphs in digraphs whose contraction is strongly connected
(Hochst\"attler). We show that all of them fit into the same framework related
to cuts in matchings. This allows us to find a counterexample to the conjecture
of Hochst\"attler and show that the conjecture of Neumann-Lara holds for all
planar graphs on at most 26 vertices. Finally, we state a new conjecture on
bipartite cubic oriented graphs, that naturally arises in this setting.Comment: 12 pages, 5 figures, 1 table. Improved expositio
Composite Rational Functions and Arithmetic Progressions
In this paper we deal with composite rational functions having zeros and
poles forming consecutive elements of an arithmetic progression. We also
correct a result published earlier related to composite rational functions
having a fixed number of zeros and poles
Universal Algebra of a Hom-Lie Algebra and group-like elements
We construct the universal enveloping algebra of a Hom-Lie algebra and endow
it with a Hom-Hopf algebra structure. We discuss group-like elements that we
see as a Hom-group integrating the initial Hom-Lie algebra
Nearly Optimal Algorithms for the Decomposition of Multivariate Rational Functions and the Extended L\"uroth's Theorem
The extended L\"uroth's Theorem says that if the transcendence degree of
\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK is 1 then there exists f \in
\KK(\underline{X}) such that \KK(\mathsf{f}_1,\dots,\mathsf{f}_m) is equal
to \KK(f). In this paper we show how to compute with a probabilistic
algorithm. We also describe a probabilistic and a deterministic algorithm for
the decomposition of multivariate rational functions. The probabilistic
algorithms proposed in this paper are softly optimal when is fixed and
tends to infinity. We also give an indecomposability test based on gcd
computations and Newton's polytope. In the last section, we show that we get a
polynomial time algorithm, with a minor modification in the exponential time
decomposition algorithm proposed by Gutierez-Rubio-Sevilla in 2001
Two for the Price of One: Lifting Separation Logic Assertions
Recently, data abstraction has been studied in the context of separation
logic, with noticeable practical successes: the developed logics have enabled
clean proofs of tricky challenging programs, such as subject-observer patterns,
and they have become the basis of efficient verification tools for Java
(jStar), C (VeriFast) and Hoare Type Theory (Ynot). In this paper, we give a
new semantic analysis of such logic-based approaches using Reynolds's
relational parametricity. The core of the analysis is our lifting theorems,
which give a sound and complete condition for when a true implication between
assertions in the standard interpretation entails that the same implication
holds in a relational interpretation. Using these theorems, we provide an
algorithm for identifying abstraction-respecting client-side proofs; the proofs
ensure that clients cannot distinguish two appropriately-related module
implementations