The generalized 4-connectivity of burnt pancake graphs

Abstract

The generalized $k$-connectivity of a graph $G$, denoted by $\kappa_k(G)$, is the minimum number of internally edge disjoint $S$-trees for any $S\subseteq V(G)$ and $|S|=k$. 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 $BP_n$ is a Cayley graph which posses many desirable properties. In this paper, we try to evaluate the reliability of $BP_n$ by investigating its generalized 4-connectivity. By introducing the notation of inclusive tree and by studying structural properties of $BP_n$, we show that $\kappa_4(BP_n)=n-1$ for $n\ge 2$, that is, for any four vertices in $BP_n$, there exist ($n-1$) internally edge disjoint trees connecting them in $BP_n$.