The Černý (1964) conjecture holds for quasi-Eulerian automata of degree one

A synchronizing automaton with n states is quasi-Eulerian of degree d if, for some positive weights on its letters, all states outside a set X of d states and one further state s receive total weight one, and every edge leaving X ends in X ∪ {s}. The Černý automata are quasi-Eulerian of degree one, and Berlinkov proved that the reset threshold of a strongly connected quasi-Eulerian automaton of degree d is at most 2^d(n-d+1)(n-1). We prove that every synchronizing quasi-Eulerian automaton of degree one has a reset word of length at most (n-1)^2. So the Černý (1964) conjecture holds for this class, and the bound is attained by the Černý automata. Moreover, the same bound holds whenever some positive, possibly irrational, letter weights have a positive stationary vector that is constant on all states but one; for this hypothesis Berlinkov's bound was 2n(n-1). For degree two we prove the bound (n-1)^2 for every automaton that is not strongly connected, for every automaton in which the two exceptional states form a block of a congruence, and, for binary strongly connected automata, in 37 of the 49 possible configurations of the two exceptional states, which include all configurations of 9 of the 14 possible stationary vectors. The main new ingredients are a bound on the length of a word that moves the exceptional state across a set, and a pairing of such moves.