Balanced generic circuits without long paths

We call a graph G = ( V , E ) a ( k , ℓ ) -circuit if | E | = k | V | − ℓ + 1 and every X ⊂ V with 2 ≤ | X | ≤ | V | − 1 induces at most k | X | − ℓ edges. A ( 2 , 3 ) -circuit is also called a generic circuit. We say that a graph is balanced if the difference between its maximum and minimum degrees is at most 1. Graver et al. asked in Graver et al. (1993) [7] whether a balanced generic circuit always admits a decomposition into two disjoint Hamiltonian paths. We show that this does not hold, moreover there are balanced ( k , k + 1 ) -circuits for all k ≥ 2 which do not contain any Hamiltonian path nor a path longer than | V | λ for λ > log 8 log 9 ≃ 0 , 9464 .