Degree sequence and supereulerian graphs

A sequence d = (d₁, d₂, ..., d_n) is graphic if there is a simple graph G with degree sequence d, and such a graph G is called a realization of d. A graphic sequence d is line-hamiltonian if d has a realization G such that L (G) is hamiltonian, and is supereulerian if d has a realization G with a spanning eulerian subgraph. In this paper, it is proved that a nonincreasing graphic sequence d = (d₁, d₂, ..., d_n) has a supereulerian realization if and only if d_n ≥ 2 and that d is line-hamiltonian if and only if either d₁ = n - 1, or ∑_{di = 1} d_i ≤ ∑_{dj ≥ 2} (d_j - 2).