24 research outputs found
Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles
Given a simple graph , a subset of is called a triangle cover if
it intersects each triangle of . Let and denote the
maximum number of pairwise edge-disjoint triangles in and the minimum
cardinality of a triangle cover of , respectively. Tuza conjectured in 1981
that holds for every graph . In this paper, using a
hypergraph approach, we design polynomial-time combinatorial algorithms for
finding small triangle covers. These algorithms imply new sufficient conditions
for Tuza's conjecture on covering and packing triangles. More precisely,
suppose that the set of triangles covers all edges in . We
show that a triangle cover of with cardinality at most can be
found in polynomial time if one of the following conditions is satisfied: (i)
, (ii) , (iii)
.
Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs,
Combinatorial algorithm
Out-of-equilibrium phonons in gated superconducting switches
Recent experiments have suggested that superconductivity in metallic nanowires can be suppressed by the application of modest gate voltages. The source of this gate action has been debated and either attributed to an electric-field effect or to small leakage currents. Here we show that the suppression of superconductivity in titanium nitride nanowires on silicon substrates does not depend on the presence or absence of an electric field at the nanowire, but requires a current of high-energy electrons. The suppression is most efficient when electrons are injected into the nanowire, but similar results are obtained when electrons are passed between two remote electrodes. This is explained by the decay of high-energy electrons into phonons, which propagate through the substrate and affect superconductivity in the nanowire by generating quasiparticles. By studying the switching probability distribution of the nanowire, we also show that high-energy electron emission leads to a much broader phonon energy distribution compared with the case where superconductivity is suppressed by Joule heating near the nanowire
Control over epitaxy and the role of the InAs/Al interface in hybrid two-dimensional electron gas systems
In-situ synthesised semiconductor/superconductor hybrid structures became an
important material platform in condensed matter physics. Their development
enabled a plethora of novel quantum transport experiments with focus on Andreev
and Majorana physics. The combination of InAs and Al has become the workhorse
material and has been successfully implemented in the form of one-dimensional
structures and two-dimensional electron gases. In contrast to the
well-developed semiconductor parts of the hybrid materials, the direct effect
of the crystal nanotexture of Al films on the electron transport still remains
unclear. This is mainly due to the complex epitaxial relation between Al and
the semiconductor. We present a study of Al films on shallow InAs
two-dimensional electron gas systems grown by molecular beam epitaxy, with
focus on control of the Al crystal structure. We identify the dominant grain
types present in our Al films and show that the formation of grain boundaries
can be significantly reduced by controlled roughening of the epitaxial
interface. Finally, we demonstrate that the implemented roughening does not
negatively impact either the electron mobility of the two-dimensional electron
gas or the basic superconducting properties of the proximitized system.Comment: 12 pages, 7 figures and supplementary materia
Bounds for graph regularity and removal lemmas
We show, for any positive integer k, that there exists a graph in which any
equitable partition of its vertices into k parts has at least ck^2/\log^* k
pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute
constants. This bound is tight up to the constant c and addresses a question of
Gowers on the number of irregular pairs in Szemer\'edi's regularity lemma.
In order to gain some control over irregular pairs, another regularity lemma,
known as the strong regularity lemma, was developed by Alon, Fischer,
Krivelevich, and Szegedy. For this lemma, we prove a lower bound of
wowzer-type, which is one level higher in the Ackermann hierarchy than the
tower function, on the number of parts in the strong regularity lemma,
essentially matching the upper bound. On the other hand, for the induced graph
removal lemma, the standard application of the strong regularity lemma, we find
a different proof which yields a tower-type bound.
We also discuss bounds on several related regularity lemmas, including the
weak regularity lemma of Frieze and Kannan and the recently established regular
approximation theorem. In particular, we show that a weak partition with
approximation parameter \epsilon may require as many as
2^{\Omega(\epsilon^{-2})} parts. This is tight up to the implied constant and
solves a problem studied by Lov\'asz and Szegedy.Comment: 62 page
The Ramsey number for hypergraph cycles
Abstract. Let Cn denote the 3-uniform hypergraph loose cycle, that is the hypergraph with vertices v1,..., vn and edges v1v2v3, v3v4v5, v5v6v7,..., vnâ1vnv1. We prove that every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of Cn, where N is asymptotically equal to 5n/4. Moreover this result is (asymptotically) best possible. 1
Ramsey numbers for bipartite graphs with small bandwidth
We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that the bipartite graph is balanced
The Ramsey number for hypergraph cycles I
AbstractLet Cn denote the 3-uniform hypergraph loose cycle, that is the hypergraph with vertices v1,âŠ,vn and edges v1v2v3, v3v4v5, v5v6v7,âŠ,vn-1vnv1. We prove that every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of Cn, where N is asymptotically equal to 5n/4. Moreover this result is (asymptotically) best possible