Interval edge colorings of some products of graphs

TL;DR

This paper investigates the conditions under which various graph products preserve the property of having an interval edge coloring, extending previous results and introducing new classes of such graphs.

Contribution

It proves that certain graph products, including lexicographic, strong, and tensor products, preserve interval colorability under specified conditions.

Findings

Cartesian product of interval colorable graphs remains interval colorable.

Lexicographic and strong products are interval colorable if one factor is regular.

Tensor and strong tensor products are interval colorable when one graph is regular.

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. A graph $G$ is interval colorable, if there is an integer $t\geq 1$ for which $G$ has an interval $t$-coloring. Let $\mathfrak{N}$ be the set of all interval colorable graphs. In 2004 Kubale and Giaro showed that if $G,H\in \mathfrak{N}$, then the Cartesian product of these graphs belongs to $\mathfrak{N}$. Also, they formulated a similar problem for the lexicographic product as an open problem. In this paper we first show that if $G\in \mathfrak{N}$, then $G[nK_{1}]\in \mathfrak{N}$ for any $n\in \mathbf{N}$. Furthermore, we show that if $G,H\in \mathfrak{N}$ and $H$ is a regular graph, then strong and lexicographic products of graphs $G,H$ belong to $\mathfrak{N}$. We also prove that tensor and strong tensor products of graphs $G,H$ belong to $\mathfrak{N}$ if $G\in \mathfrak{N}$ and $H$ is a regular graph.