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Proof of a local antimagic conjecture
An antimagic labelling of a graph is a bijection
such that the sums
distinguish all vertices. A well-known conjecture of Hartsfield and Ringel
(1994) is that every connected graph other than admits an antimagic
labelling. Recently, two sets of authors (Arumugam, Premalatha, Ba\v{c}a \&
Semani\v{c}ov\'a-Fe\v{n}ov\v{c}\'ikov\'a (2017), and Bensmail, Senhaji \&
Lyngsie (2017)) independently introduced the weaker notion of a local antimagic
labelling, where only adjacent vertices must be distinguished. Both sets of
authors conjectured that any connected graph other than admits a local
antimagic labelling. We prove this latter conjecture using the probabilistic
method. Thus the parameter of local antimagic chromatic number, introduced by
Arumugam et al., is well-defined for every connected graph other than .Comment: Final version for publication in DMTCS. Changes from previous version
are formatting to journal style and correction of two minor typographical
error
Level-dynamic approach to the excited spectra of the Jahn-Teller model - kink-train lattice and 'glassy' quantum phase
The dynamics of excited phonon spectra of the Exe Jahn-Teller (hereafter, JT)
model mapped onto the generalized Calogero-Moser (gCM) gas of pseudoparticles
implies a complex interplay between nonlinearity and fluctuations of
quasiparticle trajectories. A broad crossover appears in a pseudotime
(interaction strength) between the initial oscillator region and the nonlinear
region of the kink-train lattice as a superlattice of the kink-antikink gCM
trajectories. The local nonlinear fluctuations, nuclei (droplets) of the
growing kink phase arise at the crossover, forming a new intermediate droplet
"glassy" phase as a precursor of the kink phase. The "glassy" phase is related
to a broad maximum in the entropy of the probability distributions of
pseudoparticle accelerations, or level curvatures. The kink-train lattice phase
with multiple kink-antikink collisions is stabilised by long-range correlations
when approaching a semiclassical limit. A series of bifurcations of
nearest-level spacings were recognised as signatures of pre-chaotic behaviour
at the quantum level in the kink phase. Statistical characteristics can be seen
to confirm the coexistence within all of the spectra of both regularity and
chaoticity to a varying extent (nonuniversality). Regions are observed within
which one of the phases is dominant.Comment: 10 pages, 8 figures; published in European Physical Journal B; see
also: cond-mat/050968
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