83 research outputs found
Bounds on the diameter of Cayley graphs of the symmetric group
In this paper we are concerned with the conjecture that, for any set of
generators S of the symmetric group of degree n, the word length in terms of S
of every permutation is bounded above by a polynomial of n. We prove this
conjecture for sets of generators containing a permutation fixing at least 37%
of the points.Comment: 17 pages, 6 table
On the interplay between Babai and Cerny's conjectures
Motivated by the Babai conjecture and the Cerny conjecture, we study the
reset thresholds of automata with the transition monoid equal to the full
monoid of transformations of the state set. For automata with states in
this class, we prove that the reset thresholds are upper-bounded by
and can attain the value . In addition, we study diameters
of the pair digraphs of permutation automata and construct -state
permutation automata with diameter .Comment: 21 pages version with full proof
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