A note on interval edge-colorings of graphs

TL;DR

This paper investigates the bounds on the number of colors used in interval edge-colorings of connected graphs, establishing upper limits based on graph properties such as connectivity and regularity.

Contribution

It proves new upper bounds for the number of colors in interval edge-colorings of connected graphs, improving previous bounds for regular graphs with certain size conditions.

Findings

For connected graphs, t ≤ 2n - 3.

For connected r-regular graphs with n ≥ 2r + 2, t ≤ 2n - 5.

Provides theoretical bounds on interval edge-colorings.

Abstract

An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is called an interval $t$-coloring if for each $i\in \{1,2,\ldots,t\}$ there is at least one edge of $G$ colored by $i$, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. In this paper we prove that if a connected graph $G$ with $n$ vertices admits an interval $t$-coloring, then $t\leq 2n-3$. We also show that if $G$ is a connected $r$-regular graph with $n$ vertices has an interval $t$-coloring and $n\geq 2r+2$, then this upper bound can be improved to $2n-5$.