Multigraphic degree sequences and supereulerian graphs, disjoint spanning trees

A sequence d = ( d 1 , d 2 , … , d n ) 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 = ( d 1 , d 2 , … , d n ) has a realization with a spanning eulerian subgraph if and only if either n = 1 and d 1 = 0 , or n ≥ 2 and d n ≥ 2 , and that d has a realization G such that L ( G ) is hamiltonian if and only if either d 1 ≥ n − 1 , or ∑ d i = 1 d i ≤ ∑ d j ≥ 2 ( d j − 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 d 1 = 0 , or n ≥ 2 and both d n ≥ k and ∑ i = 1 n d i ≥ 2 k ( n − 1 ) .