One-to-one disjoint path covers in digraphs with faulty edges

Let D be a digraph of order $n\ge l+1$, where l is a positive integer. Let S=$\left\{s\right\}$ and T=$\left\{t\right\}$. A set of l paths $\{P_1,P_2,\dots ,P_l\}$ of D is a one-to-one l-disjoint directed path cover (one-to-one l-DDPC for short) for S and T, if ${\bigcup }_{i=1}^{l}V\left(P_i\right)=V\left(D\right)$, each $P_i$ is an $s-t$ path and $V\left(P_i\right)\cap V\left(P_j\right)=\{s,t\}$ for $i\ne j$. If there is a one-to-one l-DDPC in D for any disjoint source set S=$\left\{s\right\}$ and sink set $T=\left\{t\right\}$, then D is one-to-one l-coverable. In this paper, we study one-to-one disjoint path covers in digraphs with faulty edges.

We first consider complete digraphs. It is proved that for sufficiently large n, ${\overleftrightarrow{K}}_n-M$ is one-to-one l-coverable if $\left|M\right|\le \lfloor (n-l-3)/2\rfloor$. Moreover, we prove that for $\left|M\right|\le \lfloor (n-l)/2\rfloor$, ${\overleftrightarrow{K}}_n-M$ is l-ordered Hamiltonian. Also, we show that when $n\ge 1600l^3$ and $\left|M\right|\le n/2-l$, ${\overleftrightarrow{K}}_n-M$ is l-linked.

We next study complete regular bipartite digraphs. It is proved that for sufficiently large n, ${\overleftrightarrow{K}}_{n,n}-M$ is one-to-one l-coverable when $\left|M\right|\le \lfloor (n-l-1)/2\rfloor$, and ${\overleftrightarrow{K}}_{n,n}-M$ is l-ordered Hamiltonian when $\left|M\right|\le \lfloor (n-l)/2\rfloor +1$. We also prove that when $n\ge 800l^3$ and $\left|M\right|\le n/2-l+1$, ${\overleftrightarrow{K}}_{n,n}-M$ is l-linked. Furthermore, we show that for odd order $n(\ge 5)$ and $\left|M\right|\le n+1$, ${\overleftrightarrow{K}}_{n,n}-M$ is one-to-one $(n-1)$-coverable.