Mutually orthogonal hamiltonian connected graphs

In this work, we concentrate on those n -vertex graphs G with n ≥ 4 and e ̄ ≤ 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 ≠ v q for q ∈ { 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.