On Sheehan's Conjecture for Graphs with Symmetry

Sheehan's Conjecture states that every hamiltonian 4‐regular graph possesses a second Hamilton cycle. In this article, we verify Sheehan's Conjecture for 4‐regular graphs of order n whose automorphism group has size at least2 n 5 if either n ¬ ≡ 0 ( modα ) for all α ∈ { 3 , 4 , 5 } , or n = α m for α ∈ { 3 , 4 , 5 } and m is either an odd prime or a power of 2. We also give lower bounds on the size of the automorphism group and degree of a regular hamiltonian graph that guarantee existence of a second Hamilton cycle.