Upper bounds on the number of colors in interval edge-colorings of graphs

An edge-coloring of a graph G with colors 1,…,t is called 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 1990, Kamalian proved that if a graph G with at least one edge has an interval t-coloring, then t≤2|V(G)|−3. In 2002, Axenovich improved this upper bound for planar graphs: if a planar graph G admits an interval t-coloring, then t≤116|V(G)|. In the same paper Axenovich suggested a conjecture that if a planar graph G has an interval t-coloring, then t≤32|V(G)|. In this paper we first prove that if a graph G has an interval t-coloring, then t≤|E(G)|+|V(G)|−12. Next, we confirm Axenovich's conjecture by showing that if a planar graph G admits an interval t-coloring, then t≤3|V(G)|−42. We also prove that if an outerplanar graph G has an interval t-coloring, then t≤|V(G)|−1. Moreover, all these upper bounds are sharp.