On k -Edge-Hamilton-Connected Line Graphs

We say that a graph $G$ is $k$-edge-Hamilton-connected if $G+M$ has a Hamilton cycle containing all edges of $M$ for any $M\subseteq \left\{xy\mid x,y\in V\left(G\right)\right\}$ with $\mid M\mid \le k$ such that $M$ is a linear forest. In 2012, Kužel et al. conjectured that every 4-connected line graph is 2-edge-Hamilton-connected, and proved that it is equivalent to Thomassen's conjecture stating that every 4-connected line graph is Hamiltonian. In this paper, we prove that for $k\ge 2$ every $\left(k+5\right)$-connected line graph is $k$-edge-Hamilton-connected.