256 research outputs found
Shortness exponents of families of graphs
AbstractKnown estimates of the maximal length of simple circuits in certain 3-connected planar graphs are surveyed and improved in several directions
Hamiltonicity in multitriangular graphs
The family of 5-valent polyhedral graphs whose faces are all triangles or 3s-gons, s ≥ 9, is shown to contain non-hamiltonian graphs and to have a shortness exponent smaller than one
Shortness coefficient of cyclically 4-edge-connected cubic graphs
Grünbaum and Malkevitch proved that the shortness coefficient of cyclically 4-edge-connected cubic planar graphs is at most 76/77. Recently, this was improved to 359/366 (< 52/53) and the question was raised whether this can be strengthened to 41/42, a natural bound inferred from one of the Faulkner-Younger graphs. We prove that the shortness coefficient of cyclically 4-edge-connected cubic planar graphs is at most 37/38 and that we also get the same value for cyclically 4-edge-connected cubic graphs of genus g for any prescribed genus g ≥ 0. We also show that 45/46 is an upper bound for the shortness coefficient of cyclically 4-edge-connected cubic graphs of genus g with face lengths bounded above by some constant larger than 22 for any prescribed g ≥ 0
Towards a fully automated computation of RG-functions for the 3- O(N) vector model: Parametrizing amplitudes
Within the framework of field-theoretical description of second-order phase
transitions via the 3-dimensional O(N) vector model, accurate predictions for
critical exponents can be obtained from (resummation of) the perturbative
series of Renormalization-Group functions, which are in turn derived
--following Parisi's approach-- from the expansions of appropriate field
correlators evaluated at zero external momenta.
Such a technique was fully exploited 30 years ago in two seminal works of
Baker, Nickel, Green and Meiron, which lead to the knowledge of the
-function up to the 6-loop level; they succeeded in obtaining a precise
numerical evaluation of all needed Feynman amplitudes in momentum space by
lowering the dimensionalities of each integration with a cleverly arranged set
of computational simplifications. In fact, extending this computation is not
straightforward, due both to the factorial proliferation of relevant diagrams
and the increasing dimensionality of their associated integrals; in any case,
this task can be reasonably carried on only in the framework of an automated
environment.
On the road towards the creation of such an environment, we here show how a
strategy closely inspired by that of Nickel and coworkers can be stated in
algorithmic form, and successfully implemented on the computer. As an
application, we plot the minimized distributions of residual integrations for
the sets of diagrams needed to obtain RG-functions to the full 7-loop level;
they represent a good evaluation of the computational effort which will be
required to improve the currently available estimates of critical exponents.Comment: 54 pages, 17 figures and 4 table
Bipartite regular graphs and shortness parameters
By constructing sequences of non-Hamiltonian graphs it is proved that (1) for k ⩾ 4, the class of k-connected k-valent bipartite graphs has shortness exponent less than one and (2) the class of cyclically 4-edge-connected trivalent bipartite graphs has shortness coefficient less than one
On cubic polyhedral graphs with prescribed adjacency properties of their faces
AbstractWe consider classes of cubic polyhedral graphs whose non-q-gonal faces are adjacent to q-gonal faces only. Structural properties of some classes of such graphs are described. For q = 5 we show that all the graphs in this class are cyclically 4-edge-connected. Some cyclically 4edge-connected and cyclically 5-edge-connected non-Hamiltonian members from this class are presented
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