  The anti-Ramsey number, $AR(n,G)$, for a graph $G$ and an integer
$n\geq|V(G)|$, is defined to be the minimal integer $r$ such that in any
edge-colouring of $K_n$ by at least $r$ colours there is a multicoloured copy
of $G$, namely, a copy of $G$ that each of its edges has a distinct colour. In
this paper we determine, for large enough $n$, $AR(n,L\cup tP_2)$ and
$AR(n,L\cup kP_3)$ for any large enough $t$ and $k$, and a graph $L$ satisfying
some conditions. Consequently, we determine $AR(n,G)$, for large enough $n$,
where $G$ is $P_3\cup tP_2$ for any $t\geq 3$, $P_4\cup tP_2$ and $C_3\cup
tP_2$ for any $t\geq 2$, $kP_3$ for any $k\geq 3$, $tP_2\cup kP_3$ for any
$t\geq 1$, $k\geq 2$, and $P_{t+1}\cup kP_3$ for any $t\geq 3$, $k\geq 1$.
Furthermore, we obtain upper and lower bounds for $AR(n,G)$, for large enough
$n$, where $G$ is $P_{k+1}\cup tP_2$ and $C_k\cup tP_2$ for any $k\geq 4$,
$t\geq 1$.
