17,613 research outputs found
On Hamiltonian alternating cycles and paths
We undertake a study on computing Hamiltonian alternating cycles and paths on bicolored point sets. This has been an intensively studied problem, not always with a solution, when the paths and cycles are also required to be plane. In this paper, we relax the constraint on the cycles and paths from being plane to being 1-plane, and deal with the same type of questions as those for the plane case, obtaining a remarkable variety of results. For point sets in general position, our main result is that it is always possible to obtain a 1-plane Hamiltonian alternating cycle. When the point set is in convex position, we prove that every Hamiltonian alternating cycle with minimum number of crossings is 1-plane, and provide O(n) and O(n2) time algorithms for computing, respectively, Hamiltonian alternating cycles and paths with minimum number of crossings.Peer ReviewedPostprint (author's final draft
A Distributed algorithm to find Hamiltonian cycles in Gnp random graphs
In this paper, we present a distributed algorithm to find Hamiltonian cycles in random binomial graphs Gnp. The algorithm works on a synchronous distributed setting by first creating a small cycle, then covering almost all vertices in the graph with several disjoint paths, and finally patching these paths and the uncovered vertices to the cycle. Our analysis shows that, with high probability, our algorithm is able to find a Hamiltonian cycle in Gnp when p_n=omega(sqrt{log n}/n^{1/4}). Moreover, we conduct an average case complexity analysis that shows that our algorithm terminates in expected sub-linear time, namely in O(n^{3/4+epsilon}) pulses.Postprint (published version
Competing Adiabatic Thouless Pumps in Enlarged Parameter Spaces
The transfer of conserved charges through insulating matter via smooth
deformations of the Hamiltonian is known as quantum adiabatic, or Thouless,
pumping. Central to this phenomenon are Hamiltonians whose insulating gap is
controlled by a multi-dimensional (usually two-dimensional) parameter space in
which paths can be defined for adiabatic changes in the Hamiltonian, i.e.,
without closing the gap. Here, we extend the concept of Thouless pumps of band
insulators by considering a larger, three-dimensional parameter space. We show
that the connectivity of this parameter space is crucial for defining quantum
pumps, demonstrating that, as opposed to the conventional two-dimensional case,
pumped quantities depend not only on the initial and final points of
Hamiltonian evolution but also on the class of the chosen path and preserved
symmetries. As such, we distinguish the scenarios of closed/open paths of
Hamiltonian evolution, finding that different closed cycles can lead to the
pumping of different quantum numbers, and that different open paths may point
to distinct scenarios for surface physics. As explicit examples, we consider
models similar to simple models used to describe topological insulators, but
with doubled degrees of freedom compared to a minimal topological insulator
model. The extra fermionic flavors from doubling allow for extra gapping
terms/adiabatic parameters - besides the usual topological mass which preserves
the topology-protecting discrete symmetries - generating an enlarged adiabatic
parameter-space. We consider cases in one and three \emph{spatial} dimensions,
and our results in three dimensions may be realized in the context of
crystalline topological insulators, as we briefly discuss.Comment: 21 pages, 7 Figure
Approximating Longest Path
We investigate the computational hardness of approximating the longest path and the longest cycle in undirected and directed graphs on n vertices. We show that * in any expander graph, we can find (n) long paths in polynomial time. * there is an algorithm that finds a path of length (log2 L/ log log L) in any undirected graph having a path of length L, in polynomial time. * there is an algorithm that finds a path of length (log2 n/ log log n) in any Hamiltonian directed graph of constant bounded outdegree, in polynomial time. * there cannot be an algorithm finding paths of length (n ) for any constant > 0 in a Hamiltonian directed graph of bounded outdegree in polynomial time, unless P = NP. * there cannot be an algorithm finding paths of length (log2+ n), or cycles of length (log1+ n) for any constant > 0 in a Hamiltonian directed graph of constant bounded outdegree in polynomial time, unless 3-Sat can be solved in subexponential time
On almost hypohamiltonian graphs
A graph is almost hypohamiltonian (a.h.) if is non-hamiltonian, there
exists a vertex in such that is non-hamiltonian, and is
hamiltonian for every vertex in . The second author asked in [J.
Graph Theory 79 (2015) 63--81] for all orders for which a.h. graphs exist. Here
we solve this problem. To this end, we present a specialised algorithm which
generates complete sets of a.h. graphs for various orders. Furthermore, we show
that the smallest cubic a.h. graphs have order 26. We provide a lower bound for
the order of the smallest planar a.h. graph and improve the upper bound for the
order of the smallest planar a.h. graph containing a cubic vertex. We also
determine the smallest planar a.h. graphs of girth 5, both in the general and
cubic case. Finally, we extend a result of Steffen on snarks and improve two
bounds on longest paths and longest cycles in polyhedral graphs due to
Jooyandeh, McKay, {\"O}sterg{\aa}rd, Pettersson, and the second author.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1602.0717
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