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On Hamiltonian graphs arising from spatial orders of chromosomes of even number

100%

Dorninger D. , Hueber B.

2001

tom Vol. 34, nr 3

733-742

According to Bennett's model of cytogenetics the spatial order in haploid chromosome complements is based on a similarity relation which gives rise to a multigraph G which is the edge-disjoint union of two of its subgraphs G1 and G2. For even chromosome numbers n Bennett's model postulates that the order of the n chromosomes is given by a Hamiltonian circuit of G alternating in the edges of G1 and G2 o However, such a Hamiltonian circuit does not always exist. We impose a weak condition on the similarity relation and prove that under this condition the assumed Hamiltonian circuit does exist for all even chromosome numbers n < 50, which settles the case for all biological relevant species with n pairs of chromosomes. Moreover we study the structure of the graph G in respect to cycle decompositions and possible generalizations of our results.

On ring-like structures related to symmetric cryptosystems

80%

Dorninger D. , Länger H. , Mączyński M.

tom Vol. 38, nr 2

265--276

The aim of this paper is to study cryptographic systems denned as algebras (A,+alfa,+beta,p,s) of type (2,2,0,0) satisfying the axiom (x+alfa p)+beta S = x for all x is an element of A. Some standard and non-standard examples of such systems are given. In particular, we study the systems for which + alfa = + beta= + and p = s where the operation + is the addition operation of a generalized Boolean quasiring (GBQR). We investigate the structure of these algebras revealing their relation to orthomodular lattices and characterize the systems for which s (which is interpreted as coding and decoding key) commutes with all elements of A. By applying direct products to cryptographic algebras one can construct complicated cryptographic systems which may be of importance for practical use. (Then the keys are sequences whose components may be selected at random like in an XOR2 protocol.)