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

Abstract

An edge-coloring of a graph G with colors $1,\dots ,t$ 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 $t\le 2\left|V\right(G\left)\right|-3$. In 2002, Axenovich improved this upper bound for planar graphs: if a planar graph G admits an interval t-coloring, then $t\le \frac{11}{6}\left|V\right(G\left)\right|$. In the same paper Axenovich suggested a conjecture that if a planar graph G has an interval t-coloring, then $t\le \frac{3}{2}\left|V\right(G\left)\right|$. In this paper we first prove that if a graph G has an interval t-coloring, then $t\le \frac{\left|E\right(G\left)\right|+\left|V\right(G\left)\right|-1}{2}$. Next, we confirm Axenovich's conjecture by showing that if a planar graph G admits an interval t-coloring, then $t\le \frac{3\left|V\right(G\left)\right|-4}{2}$. We also prove that if an outerplanar graph G has an interval t-coloring, then $t\le \left|V\right(G\left)\right|-1$. Moreover, all these upper bounds are sharp.