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

An edge-coloring of a graph GG with colors 1,…,t1,…,t is an interval  tt-coloring   if all colors are used, and the colors of edges incident to each vertex of GG are distinct and form an interval of integers. In 1994, Asratian and Kamalian proved that if a connected graph GG admits an interval tt-coloring, then t≤(diam(G)+1)(Δ(G)−1)+1, and if GG is also bipartite, then this upper bound can be improved to t≤diam(G)(Δ(G)−1)+1, where Δ(G)Δ(G) is the maximum degree of GG and diam(G) is the diameter of GG. In this note, we show that these upper bounds cannot be significantly improved.