Hamiltonian threshold graphs

Abstract The vertices of a threshold graph G are partitioned into a clique K and an independent set I so that the neighborhoods of the vertices of I are totally ordered by inclusion. The question of whether G is hamiltonian is reduced to the case that K and I have the same size, say r , in which case the edges of K do not affect the answer and may be dropped from G , yielding a bipartite graph B . Let d 1 ≤ d 2 ≤…≤ d r and e 1 ≤ e 2 ≤…≤ e r be the degrees in B of the vertices of I and K , respectively. For each q = 0, 1,…, r −1, denote by S q the following system of inequalities: d j ⩾ j + 1, j = 1,2,…, q , e j ⩾ j + 1, j = 1, 2,…, r −1−1. Then the following conditions are equivalent: 1. (1) B is hamiltonian, 2. (2) S q holds for some q = 0, 1,…, r −1, 3. (3) S q holds for each q = 0, 1,…, r −1.