On interval edge-colorings of planar graphs

TL;DR

This paper proves a conjecture that bounds the number of colors in interval edge-colorings of planar graphs, improving previous bounds and confirming the conjecture with sharp upper limits.

Contribution

The paper confirms Axenovich's conjecture on planar graphs and establishes sharp upper bounds for interval edge-colorings of planar and outerplanar graphs.

Findings

Planar graphs with interval t-colorings satisfy t ≤ (3|V(G)| - 4)/2.

Outerplanar graphs with interval t-colorings satisfy t ≤ |V(G)| - 1.

All established bounds are proven to be sharp.

Abstract

An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an \emph{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 $t\leq 2|V(G)|-3$. In 2002, Axenovich improved this upper bound for planar graphs: if a planar graph $G$ admits an interval $t$-coloring, then $t\leq \frac{11}{6}|V(G)|$. In the same paper Axenovich suggested a conjecture that if a planar graph $G$ has an interval $t$-coloring, then $t\leq \frac{3}{2}|V(G)|$. In this paper we confirm the conjecture by showing that if a planar graph $G$ admits an interval $t$-coloring, then $t\leq \frac{3|V(G)|-4}{2}$. We also prove that if an outerplanar graph $G$ has an interval $t$-coloring, then $t\leq |V(G)|-1$. Moreover, all these upper bounds are sharp.