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=1di≤∑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 ∑i=1ndi≥2k(n−1).