Let G=(V,E) be a graph of order n. A Hamiltonian cycle of G is a cycle that contains every vertex in G. Two Hamiltonian cycles C1= and C2= of G are independent if u1=v1 and ui ≠ vi for 2 ≤ i ≤ n. A set of Hamiltonian cycles {C1, C2, , Ck} of G is mutually independent if its elements are pairwise independent. The mutually independent hamiltonicity IHC(G) of a graph G is the maximum integer k such that for any vertex u of G there are k mutually independent Hamiltonian cycles of G starting at u. For the n-dimensional burnt pancake graph Bn, this paper proved that IHC(B2)=1 and IHC(Bn)=n for n ≥ 3.

In this paper, we propose an algorithm that solves the node-to-node disjoint paths problem in n-burnt pancake graphs in polynomial-order time of n. We also give a proof of its correctness as well as the estimates of time complexity O(n3) and the maximum path length 3n+4. We conducted a computer experiment for n=2 to 100 to measure the average performance of our algorithm. The results show that the average time complexity is O(n3.0) and the maximum path length is 3n+4.

A burnt pancake graph is a variant of Cayley graphs and its topology is suitable for massively parallel systems. However, for a burnt pancake graph, there is much room for further research. Hence, in this study, we focus on n-burnt pancake graphs and propose an algorithm to obtain n disjoint paths from a source node to n destination nodes in polynomial order time of n, n being the degree of the graph. In addition, we estimate the time complexity of the algorithm and the sum of path lengths. We also give a proof of correctness of the algorithm. Moreover, we report the results of computer simulation to evaluate the average performance of the algorithm.

In this paper, we give an algorithm for the node-to-set disjoint paths problem in pancake graphs with its evaluation results. The algorithm is of polynomial order of n for an n-pancake graph. It is based on recursion and divided into two cases according to the distribution of destination nodes in classes into which all the nodes in a pancake graph are categorized. The sum of lengths of paths obtained and the time complexity of the algorithm are estimated and the average performance is evaluated based on computer simulation.

For any pair of distinct nodes in an n-pancake graph, we give an algorithm for construction of n-1 internally disjoint paths connecting the nodes in the time complexity of polynomial order of n. The length of each path obtained and the time complexity of the algorithm are estimated theoretically and verified by computer simulation.