  An edge coloring of a graph $G$ with colors $1,2,..., t$ is called an
interval $t$-coloring if for each $i\in \{1,2,...,t\}$ there is at least one
edge of $G$ colored by $i$, the colors of edges incident to any vertex of $G$
are distinct and form an interval of integers. In 1994 Asratian and Kamalian
proved that if a connected graph $G$ admits an interval $t$-coloring, then
$t\leq (d+1) (\Delta -1) +1$, and if $G$ is also bipartite, then this upper
bound can be improved to $t\leq d(\Delta -1) +1$, where $\Delta$ is the maximum
degree in $G$ and $d$ is the diameter of $G$. In this paper we show that these
upper bounds can not be significantly improved.
