Cyclically 5-edge-connected cubic planar graphs and shortness coefficients
It is shown that some classes of cyclically 5-edge-connected cubic planar graphs with only one type of face besides pentagons contain non-Hamiltonian members and have shortness coefficients less than unity.
Regular planar graphs with faces of only two types and shortness parameters
We construct a small non-Hamiltonian 3-connected trivalent planar graph whose faces are all 4-gons or 7-gons and show that the shortness coefficient of the class of such graphs is less than one. Then, by transforming non-Hamiltonian trivalent graphs into regular graphs of valency four or five, we obtain our main results, as follows. We show first that the class of 3-connected r-valent planar graphs whose faces are of only two types, triangles and q-gons, contains non-Hamiltonian members and has a shortness exponent less than one when r = 4, for all q ≧ 12. Under the extra restriction that, among graphs of connectivity three, only those with maximum cyclic edge-connectivity are to be considered, we prove the same result also when r = 4, for q = 20, and when r = 5, for all q ≧ 14 except multiples of three.
Extensions of the poisson bracket to differential forms and multi-vector fields
Abstract The Poisson bracket defined originally on the smooth function algebra of a Poisson manifold is extended to the space of all co-exact forms of this manifold. For the extended bracket analogues of the basic constructions and formulae of the standard hamiltonian formalism are given. The Poisson bracket is extended also, in a dual way, to the space of all co-exact multi-vector fields. Finally, we define the graded Lie algebra homomorphisms connecting these extended brackets and their “differentials” as well. The method used is based on the “unification” techniques introduced by the second author.