7,919 research outputs found
Cuts in matchings of 3-connected cubic graphs
We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette,
Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and
on even graphs in digraphs whose contraction is strongly connected
(Hochst\"attler). We show that all of them fit into the same framework related
to cuts in matchings. This allows us to find a counterexample to the conjecture
of Hochst\"attler and show that the conjecture of Neumann-Lara holds for all
planar graphs on at most 26 vertices. Finally, we state a new conjecture on
bipartite cubic oriented graphs, that naturally arises in this setting.Comment: 12 pages, 5 figures, 1 table. Improved expositio
Geometry of tropical moduli spaces and linkage of graphs
We prove the following "linkage" theorem: two p-regular graphs of the same
genus can be obtained from one another by a finite alternating sequence of
one-edge-contractions; moreover this preserves 3-edge-connectivity. We use the
linkage theorem to prove that various moduli spaces of tropical curves are
connected through codimension one.Comment: Final version incorporating the referees correction
Quantum Hall Ground States, Binary Invariants, and Regular Graphs
Extracting meaningful physical information out of a many-body wavefunction is
often impractical. The polynomial nature of fractional quantum Hall (FQH)
wavefunctions, however, provides a rare opportunity for a study by virtue of
ground states alone. In this article, we investigate the general properties of
FQH ground state polynomials. It turns out that the data carried by an FQH
ground state can be essentially that of a (small) directed graph/matrix. We
establish a correspondence between FQH ground states, binary invariants and
regular graphs and briefly introduce all the necessary concepts. Utilizing
methods from invariant theory and graph theory, we will then take a fresh look
on physical properties of interest, e.g. squeezing properties, clustering
properties, etc. Our methodology allows us to `unify' almost all of the
previously constructed FQH ground states in the literature as special cases of
a graph-based class of model FQH ground states, which we call \emph{accordion}
model FQH states
Hamiltonian Strongly Regular Graphs
We give a sufficient condition for a distance-regular graph to be Hamiltonian. In particular, the Petersen graph is the only connected non-Hamiltonian strongly regular graph on fewer than 99 vertices.Distance-regular graphs;Hamilton cycles JEL-code
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