Hamiltonian Properties of Hybrid-Faulty Burnt Pancake Graphs

TL;DR

This paper proves that the burnt pancake graph maintains Hamiltonian cycles and paths under certain combined edge and vertex faults, demonstrating its robustness and fault tolerance in network structures.

Contribution

It establishes the fault-tolerant Hamiltonian properties of burnt pancake graphs under hybrid faults, with tight bounds for Hamiltonian cycles and paths.

Findings

BP_n - F contains a Hamiltonian cycle for |F| ≤ n-2

BP_n - F contains a Hamiltonian path for |F| ≤ n-3

Results are optimal and bounds are tight

Abstract

We investigate the combined occurrence of edge faults and vertex faults in the burnt pancake graph (\( BP_n \)). In this paper, we prove that \( BP_n - F \), where \( F \) includes pairs of end-vertices of matching edges and fault-tolerant edges, contains a Hamiltonian cycle when \( |F| \leq n-2 \) and a Hamiltonian path when \( |F| \leq n-3 \). This establishes that \( BP_n \) is \((n-2)\)-hybrid fault Hamiltonian and \((n-3)\)-hybrid fault Hamiltonian connected for \( n \geq 3 \). These results are demonstrated to be optimal under the given conditions, with all bounds shown to be tight.