Balanced generic circuits without long paths

We call a graph G=(V,E)G=(V,E) a (k,ℓ)(k,ℓ)-circuit   if |E|=k|V|−ℓ+1|E|=k|V|−ℓ+1 and every X⊂VX⊂V with 2≤|X|≤|V|−12≤|X|≤|V|−1 induces at most k|X|−ℓk|X|−ℓ edges. A (2,3)(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)(k,k+1)-circuits for all k≥2k≥2 which do not contain any Hamiltonian path nor a path longer than |V|λ|V|λ for λ>log8log9≃0,9464.