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 , … , t is an interval t -coloring if all colors are used, and the colors of edges incident to each 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 ≤ ( diam ( G ) + 1 ) ( Δ ( G ) − 1 ) + 1 , and if G is also bipartite, then this upper bound can be improved to t ≤ diam ( G ) ( Δ ( G ) − 1 ) + 1 , where Δ ( G ) is the maximum degree of G and diam ( G ) is the diameter of G . In this note, we show that these upper bounds cannot be significantly improved.