We will say that the permutations f 1 , ..., f n are an -solution of an equation if the normalized Hamming distance between its l.h.p. and r.h.p. is ≤ . We give a sufficient conditions when near to an -solution exists an exact solution and some examples when there does not exist such a solution.

We give a definition of weakly sofic groups (w-sofic groups). Our definition is a rather natural extension of the definition of sofic groups where instead of the Hamming metric on symmetric groups we use general bi-invariant metrics on finite groups. The existence of non-w-sofic groups is equivalent to some conjecture about profinite topology on free groups.

Let G = (V, E) be a graph of order n and let 1 ≤ k < n be an integer. The k-token graph of G is the graph whose vertices are all the k-subsets of V, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this paper we characterize precisely, for each value of k, which graphs have a regular k-token graph and which connected graphs have a planar k-token graph.

In this note we show that the token graphs of fan graphs are Hamiltonian. This result provides another proof of the Hamiltonicity of Johnson graphs and also extends previous results obtained by Mirajkar and Priyanka on the token graphs of wheel graphs.

No abstract