On mutually independent hamiltonian paths

Abstract

Let P 1 = v 1 , v 2 , v 3 , . . . , v n and P 2 = u 1 , u 2 , u 3 , . . . , u n be two hamiltonian paths of G. We say that P 1 and P 2 are independent if u 1 = v 1 , u n = v n , and u i = v i for 1 < i < n. We say a set of hamiltonian paths P 1 , P 2 , . . . , P s of G between two distinct vertices are mutually independent if any two distinct paths in the set are independent. We use n to denote the number of vertices and use e to denote the number of edges in graph G. Moreover, we useē to denote the number of edges in the complement of G. Suppose that G is a graph with e ≤ n − 4 and n ≥ 4. We prove that there are at least n − 2 −ē mutually independent hamiltonian paths between any pair of distinct vertices of G except n = 5 andē = 1. Assume that G is a graph with the degree sum of any two non-adjacent vertices being at least n + 2. Let u and v be any two distinct vertices of G. We prove that there are deg G (u) + deg G (v) − n mutually independent hamiltonian paths between u and v if (u, v) ∈ E(G) and there are deg G (u) + deg G (v) − n + 2 mutually independent hamiltonian paths between u and v if otherwise. Definitions and notationFor the graph definition and notation we follow u, v) is an unordered pair of V }. We say that V is the vertex set and E is the * Corresponding author.

Securing Exposition with Parallelism in Spider-Web Networks

“…Two Hamiltonian paths P 1 =(u 1 , u 2 , …, u n(G) ) and (G) , and u i ≠ v i for every 1<i<n(G). A set of Hamiltonian paths [8], {P 1 ,P 2 , …, P k }, of G from u to v are mutually independent if any two distinct paths in the set are independent from u to v. It was found that SW(m,n) has the performance of at least two mutually independent Hamiltonian paths between any pair of bipartite nodes.…”

Networking scalable parallelism, integrity in knowledge economy &#x2014; Civilization from shuyuans toward ubiquitous university cities

“…A set of Hamiltonian paths, {P 1 ,P 2 ,…,P k }, of G from u to v is mutually independent if any two different Hamiltonian paths are independent from u to v. The mechanism of mutually independent Hamiltonian paths (MIHP) can be applied to parallel processing [18]. Such a feature is also considered for secret communications [19,20].…”

Ubiquitous Integrity via Network Integration and Parallelism—Sustaining Pedestrian/Bike Urbanism

“…Travelers, on both business and leisure purposes, can better enjoy tourism-related activities by using bicycles; specifically, bicyclists do not move in a closed room, but, probably along some narrow paths, watch sceneries or historical assets, in a thoroughly relaxed, enjoyed, and even information-realized, yet fully exercised mood. Hence, serving bicyclists well is expected in many cities, including Amsterdam, Atlanta, and other global cities or aerotropolis [1,2], where synergistically incorporated multimodal transportation is essential.…”