Hamiltonian Properties of Hybrid-Faulty Burnt Pancake Graphs

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.