Paired 2-disjoint path covers of burnt pancake graphs with faulty elements

Abstract: The burnt pancake graph $BP_n$ is the Cayley graph of the hyperoctahedral group using prefix reversals as generators. Let ${u,v}$ and ${x,y}$ be any two pairs of distinct vertices of $BP_n$ for $n\geq 4$. We show that there are $u-v$ and $x-y$ paths whose vertices partition the vertex set of $BP_n$ even if $BP_n$ has up to $n-4$ faulty elements. On the other hand, for every $n\ge3$ there is a set of $n-2$ faulty edges or faulty vertices for which such a fault-free disjoint path cover does not exist.