Degree sequence and supereulerian graphs

A sequence d = (d(1), d(2),...,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 = (d(1), d(2),...,d(n)) has a supereulerian realization if and only if d(n) >= 2 and that d is line-hamiltonian if and only if either d(1) = n - 1, or Sigma(di=1) d(i) <= Sigma(dj >= 2)(d(j) - 2). (C) 2007 Elsevier B.V. All rights reserved.