A note on edge degree and spanning trail containing given edges

Let $G$ be a simple graph with $ngeq4$ vertices and $d(x)+d(y)geq n+k$ for each edge $xyin E(G)$. 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.