Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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. Among them, we prove that a
1-plane Hamiltonian alternating cycle on a bicolored point set in general
position can always be obtained, and that when the point set is in convex
position, every Hamiltonian alternating cycle with minimum number of crossings
is 1-plane. Further, for point sets in convex position, we provide $O(n)$ and
$O(n^2)$ time algorithms for computing, respectively, Hamiltonian alternating
cycles and paths with minimum number of crossings.

28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol