A note on upper bounds for the maximum span in interval edge colorings of graphs

TL;DR

This paper investigates upper bounds for the maximum span in interval edge colorings of graphs, demonstrating that previously established bounds by Asratian and Kamalian are essentially tight and cannot be substantially improved.

Contribution

The authors prove that the existing upper bounds for the maximum span in interval edge colorings are close to optimal and cannot be significantly improved.

Findings

Existing bounds are nearly tight.

Upper bounds depend on diameter and maximum degree.

Bounds cannot be substantially improved.

Abstract

An edge coloring of a graph $G$ with colors $1,2,..., t$ is called an interval $t$-coloring if for each $i\in \{1,2,...,t\}$ there is at least one edge of $G$ colored by $i$, the colors of edges incident to any vertex of $G$ are 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 $t\leq (d+1) (\Delta -1) +1$, and if $G$ is also bipartite, then this upper bound can be improved to $t\leq d(\Delta -1) +1$, where $\Delta$ is the maximum degree in $G$ and $d$ is the diameter of $G$. In this paper we show that these upper bounds can not be significantly improved.