On interval edge-colorings of planar graphs
share
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≤11/6|V(G)|. In the same paper Axenovich suggested a conjecture that if a planar graph G has an interval t-coloring, then t≤3/2|V(G)|. In this paper we confirm the conjecture by showing that if a planar graph G admits an interval t-coloring, then t≤3|V(G)|-4/2. 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.
READ FULL TEXT