  In this paper we define and study a new family of graphs that generalises the
notions of line graphs and path graphs. Let $G$ be a graph with no loops but
possibly with parallel edges. An \emph{$\ell$-link} of $G$ is a walk of $G$ of
length $\ell \geqslant 0$ in which consecutive edges are different. We identify
an $\ell$-link with its reverse sequence. The \emph{$\ell$-link graph
$\mathbb{L}_\ell(G)$} of $G$ is the graph with vertices the $\ell$-links of
$G$, such that two vertices are joined by $\mu \geqslant 0$ edges in
$\mathbb{L}_\ell(G)$ if they correspond to two subsequences of each of $\mu$
$(\ell + 1)$-links of $G$.
  By revealing a recursive structure, we bound from above the chromatic number
of $\ell$-link graphs. As a corollary, for a given graph $G$ and large enough
$\ell$, $\mathbb{L}_\ell(G)$ is $3$-colourable. By investigating the shunting
of $\ell$-links in $G$, we show that the Hadwiger number of a nonempty
$\mathbb{L}_\ell(G)$ is greater or equal to that of $G$. Hadwiger's conjecture
states that the Hadwiger number of a graph is at least the chromatic number of
that graph. The conjecture has been proved by Reed and Seymour (2004) for line
graphs, and hence $1$-link graphs. We prove the conjecture for a wide class of
$\ell$-link graphs.
