Reconstruction of permutations distorted by single transposition errors

The reconstruction problem for permutations on $n$ elements from their erroneous patterns which are distorted by transpositions is presented in this paper. It is shown that for any $n \geq 3$ an unknown permutation is uniquely reconstructible from 4 distinct permutations at transposition distance at most one from the unknown permutation. The {\it transposition distance} between two permutations is defined as the least number of transpositions needed to transform one into the other. The proposed approach is based on the investigation of structural properties of a corresponding Cayley graph. In the case of at most two transposition errors it is shown that $\frac32(n-2)(n+1)$ erroneous patterns are required in order to reconstruct an unknown permutation. Similar results are obtained for two particular cases when permutations are distorted by given transpositions. These results confirm some bounds for regular graphs which are also presented in this paper.Comment: 5 pages, Report of paper presented at ISIT-200

On perfect codes in the dual­pancake graphs and complexity of congruence-classes in regular varieties

Here we define dual-pancake graphs and consider perfect codes in dual-pancake graphs. We find the existence of perfect codes in the dual-pancake graphs and consider some properties of them. In the second part we investigate complexity of congruence classes of algebras in some varieties.Мы определяем двойственные панкейк графы и рассматриваем совершенные коды на двойственных панкейк графах. Устанавливаем существование совершенных кодов на двойственных панкейк графах и рассматриваем некоторые их свойства. Во второй части мы исследуем сложность вычислений конгруэнц-классов алгебр в некоторых многообразиях

Topological properties of star graphs

AbstractOur purpose in the present paper is to investigate different topological properties of the recently introduced star graphs which are being viewed as attractive alternatives to n-cubes or hypercubes. These properties are interesting by themselves from a graph theory point of view as well as they have direct in generating vertex disjoint paths and minimal paths in a star graph. They can also be readily utilized to design routing algorithms and to compute contention and traffic congestion in networks that use star graphs as the underlying topology

Notes on the connectivity of Cayley coset digraphs

Hamidoune's connectivity results for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a certain way, then it is optimally vertex connected. The results are obtained by extending the methods used by Hamidoune. They are used to show that cycle-prefix graphs are optimally vertex connected. This implies that cycle-prefix graphs have good fault tolerance properties.Comment: 15 page

Classes of Symmetric Cayley Graphs over Finite Abelian Groups of Degrees 4 and 6

The present work is devoted to characterize the family of symmetric undirected Cayley graphs over finite Abelian groups for degrees 4 and 6.Comment: 12 pages. A previous version of some of the results in this paper where first announced at 2010 International Workshop on Optimal Interconnection Networks (IWONT 2010). It is accessible at http://upcommons.upc.edu/revistes/handle/2099/1037