Arc-disjoint hamiltonian paths in non-round decomposable local tournaments

Thomassen proved that a strong tournament $T$ has a pair of arc-disjoint Hamiltonian paths with distinct initial vertices and distinct terminal vertices if and only if $T$ is not an almost transitive tournament of odd order, where an almost transitive tournament is obtained from a transitive tournament with acyclic ordering $u_1,u_2,\dots ,u_n$ (i.e., $u_i\to u_j$ for all $1\le i<j\le n$) by reversing the arc $u_1{u}_n$. A digraph $D$ is a local tournament if for every vertex $x$ of $D$, both the out-neighbors and the in-neighbors of $x$ induce tournaments. Bang-Jensen, Guo, Gutin and Volkmann split local tournaments into three subclasses: the round decomposable; the non-round decomposable which are not tournaments; the non-round decomposable which are tournaments. In 2015, we proved that every 2-strong round decomposable local tournament has a Hamiltonian path and a Hamiltonian cycle which are arc-disjoint if and only if it is not the second power of an even cycle. In this paper, we discuss the arc-disjoint Hamiltonian paths in non-round decomposable local tournaments, and prove that every 2-strong non-round decomposable local tournament contains a pair of arc-disjoint Hamiltonian paths with distinct initial vertices and distinct terminal vertices. This result combining with the one on round decomposable local tournaments extends the above-mentioned result of Thomassen to 2-strong local tournaments.