  Let $G$ be a simple graph with $n\geq4$ vertices and $d(x)+d(y)\geq n+k$ for
each edge $xy\in E(G)$. In this work we prove that $G$ either contains a
spanning closed trail containing any given edge set $X$ if $|X|\leq k$, or $G$
is a well characterized graph. As a corollary, we show that line graphs of such
graphs are $k$-hamiltonian.
