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The structure of digraphs associated with the congruence x k y ( mod n )

Lawrence Somer, Michal Křížek (2011)

Czechoslovak Mathematical Journal

We assign to each pair of positive integers n and k 2 a digraph G ( n , k ) whose set of vertices is H = { 0 , 1 , , n - 1 } and for which there is a directed edge from a H to b H if a k b ( mod n ) . We investigate the structure of G ( n , k ) . In particular, upper bounds are given for the longest cycle in G ( n , k ) . We find subdigraphs of G ( n , k ) , called fundamental constituents of G ( n , k ) , for which all trees attached to cycle vertices are isomorphic.

The theory of dual groups

A. Mekler, G. Schlitt (1994)

Fundamenta Mathematicae

We study the L , w -theory of sequences of dual groups and give a complete classification of the L , w -elementary classes by finding simple invariants for them. We show that nonstandard models exist.

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