The generalized 4-connectivity of burnt pancake graphs

Abstract

The generalized $k$-connectivity of a graph $G$, denoted by ${\kappa }_k\left(G\right)$, is the minimum number of internally disjoint $S$-trees for any $S\subseteq V\left(G\right)$ and $\left|S\right|=k$. The generalized $k$-connectivity is a natural extension of the classical connectivity and plays a key role in applications related to the modern interconnection networks. An $n$-dimensional burnt pancake graph $B{P}_n$ is a Cayley graph which possesses many desirable properties. In this paper, we try to evaluate the reliability of $B{P}_n$ by investigating its generalized 4-connectivity. By introducing the definition of inclusive tree and by studying structural properties of $B{P}_n$, we show that ${\kappa }_4\left(B{P}_n\right)=n-1$ for $n\ge 2$, that is, for any four vertices in $B{P}_n$, there exist ($n-1$) internally disjoint trees connecting them in $B{P}_n$.