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 Care 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) (Delta(G) - 1) + 1, and if G is also bipartite, then this upper bound can be improved to t <= diam(G)(Delta(G) - 1) + 1, where Delta(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. (C) 2012 Elsevier B.V. All rights reserved.