3,581 research outputs found
Boxicity and separation dimension
A family of permutations of the vertices of a hypergraph is
called 'pairwise suitable' for if, for every pair of disjoint edges in ,
there exists a permutation in in which all the vertices in one
edge precede those in the other. The cardinality of a smallest such family of
permutations for is called the 'separation dimension' of and is denoted
by . Equivalently, is the smallest natural number so that
the vertices of can be embedded in such that any two
disjoint edges of can be separated by a hyperplane normal to one of the
axes. We show that the separation dimension of a hypergraph is equal to the
'boxicity' of the line graph of . This connection helps us in borrowing
results and techniques from the extensive literature on boxicity to study the
concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to
WG-2014. Some results proved in this paper are also present in
arXiv:1212.6756. arXiv admin note: substantial text overlap with
arXiv:1212.675
Boxicity of Series Parallel Graphs
The three well-known graph classes, planar graphs (P), series-parallel
graphs(SP) and outer planar graphs(OP) satisfy the following proper inclusion
relation: OP C SP C P. It is known that box(G) <= 3 if G belongs to P and
box(G) <= 2 if G belongs to OP. Thus it is interesting to decide whether the
maximum possible value of the boxicity of series-parallel graphs is 2 or 3. In
this paper we construct a series-parallel graph with boxicity 3, thus resolving
this question. Recently Chandran and Sivadasan showed that for any G, box(G) <=
treewidth(G)+2. They conjecture that for any k, there exists a k-tree with
boxicity k+1. (This would show that their upper bound is tight but for an
additive factor of 1, since the treewidth of any k-tree equals k.) The
series-parallel graph we construct in this paper is a 2-tree with boxicity 3
and is thus a first step towards proving their conjecture.Comment: 10 pages, 0 figure
Hadwiger Number and the Cartesian Product Of Graphs
The Hadwiger number mr(G) of a graph G is the largest integer n for which the
complete graph K_n on n vertices is a minor of G. Hadwiger conjectured that for
every graph G, mr(G) >= chi(G), where chi(G) is the chromatic number of G. In
this paper, we study the Hadwiger number of the Cartesian product G [] H of
graphs.
As the main result of this paper, we prove that mr(G_1 [] G_2) >= h\sqrt{l}(1
- o(1)) for any two graphs G_1 and G_2 with mr(G_1) = h and mr(G_2) = l. We
show that the above lower bound is asymptotically best possible. This
asymptotically settles a question of Z. Miller (1978).
As consequences of our main result, we show the following:
1. Let G be a connected graph. Let the (unique) prime factorization of G be
given by G_1 [] G_2 [] ... [] G_k. Then G satisfies Hadwiger's conjecture if k
>= 2.log(log(chi(G))) + c', where c' is a constant. This improves the
2.log(chi(G))+3 bound of Chandran and Sivadasan.
2. Let G_1 and G_2 be two graphs such that chi(G_1) >= chi(G_2) >=
c.log^{1.5}(chi(G_1)), where c is a constant. Then G_1 [] G_2 satisfies
Hadwiger's conjecture.
3. Hadwiger's conjecture is true for G^d (Cartesian product of G taken d
times) for every graph G and every d >= 2. This settles a question by Chandran
and Sivadasan (They had shown that the Hadiwger's conjecture is true for G^d if
d >= 3.)Comment: 10 pages, 2 figures, major update: lower and upper bound proofs have
been revised. The bounds are now asymptotically tigh
Weakly Turbulent MHD Waves in Compressible Low-Beta Plasmas
In this Letter, weak turbulence theory is used to investigate interactions
among Alfven waves and fast and slow magnetosonic waves in collisionless
low-beta plasmas. The wave kinetic equations are derived from the equations of
magnetohydrodynamics, and extra terms are then added to model collisionless
damping. These equations are used to provide a quantitative description of a
variety of nonlinear processes, including "parallel" and "perpendicular" energy
cascade, energy transfer between wave types, "phase mixing," and the generation
of back-scattered Alfven waves.Comment: Accepted, Physical Review Letter
Turbulence and Mixing in the Intracluster Medium
The intracluster medium (ICM) is stably stratified in the hydrodynamic sense
with the entropy increasing outwards. However, thermal conduction along
magnetic field lines fundamentally changes the stability of the ICM, leading to
the "heat-flux buoyancy instability" when and the "magnetothermal
instability" when . The ICM is thus buoyantly unstable regardless of
the signs of and . On the other hand, these
temperature-gradient-driven instabilities saturate by reorienting the magnetic
field (perpendicular to when and parallel to when ), without generating sustained convection. We show that
after an anisotropically conducting plasma reaches this nonlinearly stable
magnetic configuration, it experiences a buoyant restoring force that resists
further distortions of the magnetic field. This restoring force is analogous to
the buoyant restoring force experienced by a stably stratified adiabatic
plasma. We argue that in order for a driving mechanism (e.g, galaxy motions or
cosmic-ray buoyancy) to overcome this restoring force and generate turbulence
in the ICM, the strength of the driving must exceed a threshold, corresponding
to turbulent velocities . For weaker driving, the ICM
remains in its nonlinearly stable magnetic configuration, and turbulent mixing
is effectively absent. We discuss the implications of these findings for the
turbulent diffusion of metals and heat in the ICM.Comment: 8 pages, 2 figs., submitted to the conference proceedings of "The
Monster's Fiery Breath;" a follow up of arXiv:0901.4786 focusing on the
general mixing properties of the IC
Parker/buoyancy instabilities with anisotropic thermal conduction, cosmic rays, and arbitrary magnetic field strength
We report the results of a local stability analysis for a magnetized,
gravitationally stratified plasma containing cosmic rays. We account for
cosmic-ray diffusion and thermal conduction parallel to the magnetic field and
allow beta to take any value, where p is the plasma pressure and B is the
magnetic field strength. We take the gravitational acceleration to be in the
-z-direction and the equilibrium magnetic field to be in the y-direction, and
we derive the dispersion relation for small-amplitude instabilities and waves
in the large-|k_x| limit. We use the Routh-Hurwitz criterion to show
analytically that the necessary and sufficient criterion for stability in this
limit is n k_B dT/dz + dp_cr/dz + (1/8pi)dB^2/dz > 0, where T is the
temperature, n is the number density of thermal particles, and p_cr is the
cosmic-ray pressure. We present approximate analytical solutions for the normal
modes in the low- and high-diffusivity limits, show that they are consistent
with the derived stability criterion, and compare them to numerical results
obtained from the full, unapproximated, dispersion relation. Our results extend
earlier analyses of buoyancy instabilities in galaxy-cluster plasmas to the
beta <= 1 regime. Our results also extend earlier analyses of the Parker
instability to account for anisotropic thermal conduction, and show that the
interstellar medium is more unstable to the Parker instability than was
predicted by previous studies in which the thermal plasma was treated as
adiabatic.Comment: 36 pages, 2 figures, Accepted for publication in Ap
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