The generalized 4-connectivity of burnt pancake graphs

TL;DR

This paper investigates the generalized 4-connectivity of burnt pancake graphs, showing that for any four vertices, there are (n-1) edge-disjoint trees connecting them, which informs the network's reliability.

Contribution

The paper establishes the exact value of the generalized 4-connectivity for burnt pancake graphs, advancing understanding of their structural robustness.

Findings

-connectivity of BP_n is n-1 for n

Existence of (n-1) edge-disjoint trees for any four vertices in BP_n

Structural properties of BP_n facilitate connectivity analysis

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 $BP_n$.