The Rabin numbers of enhanced hypercubes

The ω-Rabin number $r_{\omega }\left(G\right)$ and strong ω-Rabin number $r_{\omega }^{⁎}\left(G\right)$ are two effective parameters to assess transmission latency and fault tolerance of an interconnection network G. As determining the Rabin number of a general graph is NP-complete, we consider the Rabin number of the enhanced hypercube $Q_{n,k}$ which is a variant of the hypercube $Q_n$. For $n\ge k\ge 5$, we prove that $r_{\omega }\left(Q_{n,k}\right)=r_{\omega }^{⁎}\left(Q_{n,k}\right)=d\left(Q_{n,k}\right)$ for $1\le \omega <n-\lfloor \frac{k}{2}\rfloor$; $r_{\omega }\left(Q_{n,k}\right)=r_{\omega }^{⁎}\left(Q_{n,k}\right)=d\left(Q_{n,k}\right)+1$ for $n-\lfloor \frac{k}{2}\rfloor \le \omega \le n+1$, where $d\left(Q_{n,k}\right)$ is the diameter of $Q_{n,k}$. In addition, we present algorithms to construct internally disjoint paths of length at most ${r_{\omega }}^{⁎}\left(Q_{n,k}\right)$ from a source vertex to other ω ($1\le \omega \le n+1$) destination vertices (not necessarily distinct) in $Q_{n,k}$.