Mutually orthogonal hamiltonian connected graphs

Abstract

in this work, we concentrate on those n-vertex graphs G with n >= 4 and (e) over bar <= n - 4. Let P(1) = < u(1), u(2),..., u(n)) and P(2) = (v(1), v(2),..., v(n)) be any two hamiltonian paths of G. We say that P(1) and P(2) are orthogonal if u(1) = v(1), u(n) = v(n), and u(q) not equal v(q) for q is an element of {2, n - 1}. We say that a set of hamiltonian paths {P(1), P(2),..., P(s)} of G are mutually orthogonal if any two distinct paths in the set are orthogonal. We will prove that there are at least two orthogonal hamiltonian paths of G between any two different vertices. Furthermore, we classify the cases such that there are exactly two orthogonal hamiltonian paths of G between any two different vertices. Aside from these special cases, there are at least three mutually orthogonal hamiltonian paths of G between any two different vertices. (C) 2009 Elsevier Ltd. All rights reserved.