  For a bipartite graph $G$ with parts $X$ and $Y$, an $X$-interval coloring is
a proper edge coloring of $G$ by integers such that the colors on the edges
incident to any vertex in $X$ form an interval. Denote by $\chi'_{int}(G,X)$
the minimum $k$ such that $G$ has an $X$-interval coloring with $k$ colors. The
author and Toft conjectured [Discrete Mathematics 339 (2016), 2628--2639] that
there is a polynomial $P(x)$ such that if $G$ has maximum degree at most
$\Delta$, then $\chi'_{int}(G,X) \leq P(\Delta)$. In this short note, we prove
this conjecture; in fact, we prove that a cubic polynomial suffices. We also
deduce some improved upper bounds on $\chi'_{int}(G,X)$ for bipartite graphs
with small maximum degree.
