Degree sequence and supereulerian graphs

A sequence d = (d1, d2, · · · , dn) 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 linehamiltonian if d has a realizationG 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, · · · , dn) has a supereulerian realization if and only if dn ≥ 2 and that d is line-hamiltonian if and only if either d1 = n − 1, or ∑ di=1 di ≤ ∑ dj≥2(dj − 2).