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.
Non-Hamiltonian simple 3-polytopes having just two types of faces
It is shown that for every value of an integer k, k>=11, there exist 3-valent 3-connected planar graphs having just two types of faces, pentagons and k-gons, and which are non- Hamiltonian. Moreover, there exist @[email protected](k) > 0, for these values of k, and sequences (G"n)^~"n"="1 of the said graphs for which V(G"n)->~ and the size of a largest circuit of G"n is at most ([email protected])V(G"n); similar result for the size of a largest path in such graphs is established for all k, k>=11, except possibly for k = 14, 17, 22 and k = 5m+ 5 for all m>=2.
How to do a p-descent on an elliptic curve
In this paper, we describe an algorithm that reduces the computation of the (full) p-Selmer group of an elliptic curve E over a number field to standard number field computations such as determining the (p-torsion of) the S-class group and a basis of the S-units modulo pth powers for a suitable set S of primes. In particular, we give a result reducing this set S of 'bad primes' to a very small set, which in many cases only contains the primes above p. As of today, this provides a feasible algorithm for performing a full 3-descent on an elliptic curve over Q, but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of E[p] is favorable, simplifications are possible and p-descents for larger p are accessible even today. To demonstrate how the method works, several worked examples are included.