The　$k$-ary　$n$-cube　$Q_{n}^{k}$　is　one　of　the　most　popular　interconnection　networks　engaged　as　the　underlying　topology　of　data　center　networks,　on-chip　networks,　and　parallel　and　distributed　systems.　Due　to　the　increasing　probability　of　faulty　edges　in　large-scale　networks　and　extensive　applications　of　the　Hamiltonian　path,　it　becomes　more　and　more　critical　to　investigate　the　fault　tolerability　of　interconnection　networks　when　embedding　the　Hamiltonian　path.　However,　since　the　existing　edge　fault　models　in　the　current　literature　only　focus　on　the　entire　status　of　faulty　edges　while　ignoring　the　important　information　in　the　edge　dimensions,　their　fault　tolerability　is　narrowed　to　a　minimal　scope.　This　paper　first　proposes　the　concept　of　the　partitioned　fault　model　to　achieve　an　exponential　scale　of　fault　tolerance.　Based　on　this　model,　we　put　forward　two　novel　indicators　for　the　bipartite　networks　(including　$Q^{k}_{n}$　with　even　$k$),　named　partition-edge　fault-tolerant　Hamiltonian　laceability　and　partition-edge　fault-tolerant　hyper-Hamiltonian　laceability.　Then,　we　exploit　these　metrics　to　explore　the　existence　of　Hamiltonian　paths　and　unpaired　2-disjoint　path　cover　in　$k$-ary　$n$-cubes　with　large-scale　faulty　edges.　Moreover,　we　prove　that　all　these　results　are　optimal　in　the　sense　that　the　number　of　edge　faults　tolerated　has　attended　to　the　best　upper　bound.　Our　approach　is　the　first　time　that　can　still　embed　a　Hamiltonian　path　and　an　unpaired　2-disjoint　path　cover　into　the　$k$-ary　$n$-cube　even　if　the　faulty　edges　grow　exponentially.　IEEE