Degree sequence and supereulerian graphs

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