  We obtain several lower bounds on the $\textsf{Max-Cut}$ of $d$-degenerate
$H$-free graphs. Let $f(m,d,H)$ denote the smallest $\textsf{Max-Cut}$ of an
$H$-free $d$-degenerate graph on $m$ edges. We show that $f(m,d,K_r)\ge
\left(\frac{1}{2} + d^{-1+\Omega(r^{-1})}\right)m$, generalizing a recent work
of Carlson, Kolla, and Trevisan. We also give bounds on $f(m,d,H)$ when $H$ is
a cycle, odd wheel, or a complete bipartite graph with at most 4 vertices on
one side. We also show stronger bounds on $f(m,d,K_r)$ assuming a conjecture of
Alon, Bollabas, Krivelevich, and Sudakov (2003). We conjecture that
$f(m,d,K_r)= \left( \frac{1}{2} + \Theta_r(d^{-1/2}) \right)m$ for every $r\ge
3$, and show that this conjecture implies the ABKS conjecture.
