Balanced generic circuits without long paths

We call a graph G = (V,E) a (k, `)-circuit if |E| = k|V | − ` + 1 and every X ⊂ V with 2 ≤ |X| ≤ |V | − 1 satisfies i(X) ≤ k|X| − `. 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 degree is at most 1. J. Graver, B. Servatius and H. Servatius asked in [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.