Paths in Folded Hypercubes With Any Pair of Faulty Vertices

When a vertex in the network is subjected to an external attack, such as a hacker attack, it is highly likely to infect one of its neighbors, resulting in both adjacent vertices being unable to operate normally in the network. Hence, it is crucial to ensure fault-tolerance capability when dealing with a pair of adjacent faulty vertices in the network, as it is essential for maintaining system reliability. The folded hypercube, denoted as $FQ_{n}$ , is a recognized variant of the hypercube architecture. It is formed by connecting every pair of vertices with complementary addresses. Notably, for any odd integer n, the $FQ_{n}$ structure exhibits bipartite characteristics. Consider two adjacent faulty vertices, $f_{1 }$ and $f_{2 }$ , within $FQ_{n}$ . Let u and v be any two fault-free vertices in ${FQ_{n}}-\{{f_{1 }},~{f_{2 }}\}{}$ . In previous studies, we investigated the embedding of cycles of various lengths in ${FQ_{n}}-\left \{{{{f_{1 }},~{f_{2 }}}}\right \}{}$ . Furthermore, we explored any fault-free edge in ${FQ_{n}}-\left \{{{{f_{1 }},~{f_{2 }}}}\right \}{}$ could be part of cycles of different lengths. Based on these findings on cycle embedding properties, we were motivated to consider whether, under the same fault-tolerant conditions, paths of various lengths could also be embedded in ${FQ_{n}}-\left \{{{{f_{1 }},~{f_{2 }}}}\right \}{}$ . Therefore, the main results examined in this paper include: 1) for every odd $n{\geq }3$ , ${FQ_{n}}-\{{f_{1 }},~{f_{2 }}\}{}$ contains a fault-free path $P[u,v]$ of every length l in the range ${d_{FQ_{n}}}\left ({{u,v}}\right){\leq }l{\leq }{2^{n}}-3$ , where $l-{d_{FQ_{n}}}\left ({{u,v}}\right)$ is even; and 2) for every even $n{\geq }4$ , ${FQ_{n}}-\{{f_{1 }},~{f_{2 }}\}{}$ contains a fault-free path $P[u,v]$ of every length l in the range $n-1{\leq }l\leq {2^{n}}-3$ . The variety of path lengths embedded in this paper represents the optimal scenario under the fault-tolerant condition.