The minimum number of Hamilton cycles in a Hamiltonian threshold graph of a prescribed order

Abstract

We prove that the minimum number of Hamilton cycles in a Hamiltonian threshold graph of order n is 2 ⌊ ( n − 3 ) ∕ 2 ⌋ and this minimum number is attained uniquely by the graph with degree sequence n goodbreakinfix− 1 goodbreakinfix, n goodbreakinfix− 1 goodbreakinfix, n goodbreakinfix− 2 , … , ⌈ n ∕ 2 ⌉ goodbreakinfix, ⌈ n ∕ 2 ⌉ , … , 3,2 of n goodbreakinfix− 2 distinct degrees. This graph is also the unique graph of minimum size among all Hamiltonian threshold graphs of order n.