Multigraphic degree sequences and supereulerian graphs, disjoint spanning trees

A sequence d = (d1, d2, . . . , dn) is multigraphic if there is a multigraph G with degree sequence d, and such a graph G is called a realization of d. In this paper, we prove that a nonincreasing multigraphic sequence d = (d1, d2, . . . , dn) has a realization with a spanning eulerian subgraph if and only if either n = 1 and d1 = 0, or n ≥ 2 and dn ≥ 2, and that d has a realization G such that L(G) is hamiltonian if and only if either d1 ≥ n− 1, or  di=1 di ≤  dj≥2 (dj−2). Also, we prove that, for a positive integer k, d has a realization with k edge-disjoint spanning trees if and only if either both n = 1 and d1 = 0, or n ≥ 2 and both dn ≥ k and n i=1 di ≥ 2k(n − 1). © 2011 Elsevier Ltd. All rights reserved.