Motivated by a conjecture of Liang [Y.-C. Liang. {\em Anti-magic labeling of
graphs}. PhD thesis, National Sun Yat-sen University, 2013.], we introduce a
restricted path packing problem in bipartite graphs that we call a
$\mathtt{V}$-free $2$-matching. We verify the conjecture through a weakening of
the hypergraph matching problem. We close the paper by showing that it is
NP-complete to decide whether one of the color classes of a bipartite graph can
be covered by a $\mathtt{V}$-free $2$-matching

Bernáth, Attila