A note on upper bounds for the maximum span in interval edge colorings of graphs

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.