  The anti-Ramsey number, $ar(G, H)$ is the minimum integer $k$ such that in
any edge colouring of $G$ with $k$ colours there is a rainbow subgraph
isomorphic to $H$, i.e., a copy of $H$ with each of its edges assigned a
different colour. The notion was introduced by Erd{\"{o}}s and Simonovits in
1973. Since then the parameter has been studied extensively in combinatorics,
also the particular case when $H$ is a star graph. Recently this case received
the attention of researchers from the algorithm community because of its
applications in interface modelling of wireless networks. To the algorithm
community, the problem is known as maximum edge $q$-colouring problem.
  In this paper, we study the maximum edge $2$-colouring problem from the
approximation algorithm point of view. The case $q=2$ is particularly
interesting due to its application in real-life problems. Algorithmically, this
problem is known to be NP-hard for $q\ge 2$. For the case of $q=2$, it is also
known that no polynomial-time algorithm can approximate to a factor less than
$3/2$ assuming the unique games conjecture. Feng et al. showed a
$2$-approximation algorithm for this problem. Later Adamaszek and Popa
presented a $5/3$-approximation algorithm with the additional assumption that
the input graph has a perfect matching. Note that the obvious but the only
known algorithm issues different colours to the edges of a maximum matching
(say $M$) and different colours to the connected components of $G \setminus M$.
In this article, we give a new analysis of the aforementioned algorithm leading
to an improved approximation bound for triangle-free graphs with perfect
matching. We also show a new lower bound when the input graph is triangle-free.
The contribution of the paper is a completely new, deeper and closer analysis
of how the optimum achieves a higher number of colours than the matching based
algorithm, mentioned above.
