  The melting phase diagram of a double-stranded DNA in poor solvent is studied
using the pruned and enriched Rosenbluth method on a simple cubic lattice. As
the solvent quality is changed from good to poor, there is a non-monotonic
change in the melting temperature. First-order melting transition, as in good
solvent, gives way to continuous transition and then to further broadened
transitions where the order parameter smoothly becomes zero for sufficiently
poor solvent. This change in the melting behavior is accompanied by a
continuously varying critical exponent along the melting curve, hinting at a
non-universal nature of the melting transition. Further, we show that an
unbound phase can be achieved just by changing the solvent quality.
Importantly, our results conform to the experimental findings qualitatively.
