25,601 research outputs found
On matchings and factors of graphs /
In Section 1, we recall the historical sketch of matching and factor theory of graphs, and also introduce some necessary definitions and notation. In Section 2, we present a sufficient condition for the existence of a (g, f)-factor in graphs with the odd-cycle property, which is simpler than that of Lovasz\u27s (g, f)-Factor Theorem. From this, we derive some further results, and we show that (a) every r-regular graph G with the odd-cycle property has a k-factor, where 0 ≤ k ≤ r and k|V(G)| ≡ 0 (mod 2), (b) every graph G with the strong odd-cycle property with k|V(G)|≡ 0 (mod 2) is k-factorable if and only if G is a km-regular graph for some m ≥ 1, and (c) every regular graph of even order with the strong odd-cycle property is of the second class (i.e. the edge chromatic number is Δ). Chvátal [26] presented the following two conjectures that (1) a graph G has a 2-factor if tough(G) ≥ 3/2, and (2) a graph G has a k-factor if k|V(G)| ≡ 0 (mod 2) and tough(G) ≥ k. Enomoto et.al. [32] proved the second conjecture. They also proved the sharpness of the bound on tough(G) that guarantees the existence of a k-factor. This implies that the first conjecture is false. In Section 3, we show that the result of the second conjecture can be improved in some sense, and the first conjecture is also true if the graph considered has the odd-cycle property. Anderson [3] stated that a graph G of even order has a 1-factor if bind(G) ≥ 4/3, and Katerinis and Woodall [48] proved that a graph G of order n has a k-factor if bind(G) ˃ (2k -I)(n - 1)/(k(n - 2) + 3), where k ≥ 2, n ≥ 4k - 6 and kn ≡ 0 (mod 2). In Section 4, we shall present some similar conditions for the existence of [a, b]-factors. In Section 5, we study the existence of [a, b]-parity-factors in a graph, among which we extend some known theorems from 1-factors to {1, 3, ... , 2n - 1}-factors, or from k-factors to [a, b]-parity-factors. Also, extending Petersen\u27s 2-Factorization Theorem, we proved that a graph is [2a, 2b]-even-factorable if and only if it is a [2na, 2nb]-even-graph for some n ≥ 1. Plummer showed that (a) (in [58]) every graph G of even order is k-extendable if tough(G) ˃ k, and (b) (in [59]) every (2k+1)-connected graph G is k-extendable if G is K1,3-free, respectively. In Section 6, we give a counterpart of the former in terms of binding number, and extend the latter from K1,3-free graphs to K1,n-free graphs. Furthermore, we present a result toward the problem, posed by Saito [61] and Plummer [60], of characterizing the graphs that are maximal k-extendable
Color/kinematics duality for general abelian orbifolds of N=4 super Yang-Mills theory
To explore color/kinematics duality for general representations of the gauge
group we formulate the duality for general abelian orbifolds of the SU(N), N=4
super Yang-Mills theory in four dimensions, which have fields in the
bi-fundamental representation, and use it to construct explicitly complete
four-vector and four-scalar amplitudes at one loop. For fixed number of
supercharges, graph-organized L-loop n-point integrands of all orbifold
theories are given in terms of a fixed set of polynomials labeled by L
representations of the orbifold group. In contrast to the standard
duality-satisfying presentation of amplitudes of the N=4 super Yang-Mills
theory, each graph may appear several times with different internal states. The
color and R-charge flow provide a way to deform the amplitudes of orbifold
theories to those of more general quiver gauge theories which do not
necessarily exhibit color/kinematics duality on their own. Based on the
organization of amplitudes required by the duality between color and kinematics
in orbifold theories we show how the amplitudes of certain non-factorized
matter-coupled supergravity theories can be found through a double-copy
construction. We also carry out a comprehensive search for theories with fields
solely in the adjoint representation of the gauge group and amplitudes
exhibiting color/kinematics duality for all external states and find an
interesting relation between supersymmetry and existence of the duality.Comment: 51 pages, 7 figures, PDFLaTex, typos fixed, minor changes in section
3.
Scattering amplitudes in YM and GR as minimal model brackets and their recursive characterization
Attached to both Yang-Mills and General Relativity about Minkowski spacetime
are distinguished gauge independent objects known as the on-shell tree
scattering amplitudes. We reinterpret and rigorously construct them as
minimal model brackets. This is based on formulating YM and GR as
differential graded Lie algebras. Their minimal model brackets are then given
by a sum of trivalent (cubic) Feynman tree graphs. The amplitudes are gauge
independent when all internal lines are off-shell, not merely up to
isomorphism, and we include a homological algebra proof of this fact. Using the
homological perturbation lemma, we construct homotopies (propagators) that are
optimal in bringing out the factorization of the residues of the amplitudes.
Using a variant of Hartogs extension for singular varieties, we give a rigorous
account of a recursive characterization of the amplitudes via their residues
independent of their original definition in terms of Feynman graphs (this does
neither involve so-called BCFW shifts nor conditions at infinity under such
shifts). Roughly, the amplitude with legs is the unique section of a sheaf
on a variety of complex momenta whose residues along a finite list of
irreducible codimension one subvarieties (prime divisors) factor into
amplitudes with less than legs. The sheaf is a direct sum of rank one
sheaves labeled by helicity signs. To emphasize that amplitudes are robust
objects, we give a succinct list of properties that suffice for a dgLa so as to
produce the YM and GR amplitudes respectively.Comment: 51 page
Representation of Cyclotomic Fields and Their Subfields
Let \K be a finite extension of a characteristic zero field \F. We say
that the pair of matrices over \F represents \K if \K
\cong \F[A]/ where \F[A] denotes the smallest subalgebra of M_n(\F)
containing and is an ideal in \F[A] generated by . In
particular, is said to represent the field \K if there exists an
irreducible polynomial q(x)\in \F[x] which divides the minimal polynomial of
and \K \cong \F[A]/. In this paper, we identify the smallest
circulant-matrix representation for any subfield of a cyclotomic field.
Furthermore, if is any prime and \K is a subfield of the -th
cyclotomic field, then we obtain a zero-one circulant matrix of size
such that (A,\J) represents \K, where \J is the matrix with
all entries 1. In case, the integer has at most two distinct prime factors,
we find the smallest 0-1 companion-matrix that represents the -th cyclotomic
field. We also find bounds on the size of such companion matrices when has
more than two prime factors.Comment: 17 page
Exact, convergent periodic-orbit expansions of individual energy eigenvalues of regular quantum graphs
We present exact, explicit, convergent periodic-orbit expansions for
individual energy levels of regular quantum graphs. One simple application is
the energy levels of a particle in a piecewise constant potential. Since the
classical ray trajectories (including ray splitting) in such systems are
strongly chaotic, this result provides the first explicit quantization of a
classically chaotic system.Comment: 25 pages, 5 figure
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