Interval edge-colorings of composition of graphs

TL;DR

This paper investigates the conditions under which the composition of graphs retains the interval colorability property, extending previous results to broader classes of graphs and specific interval colorings.

Contribution

It proves that the composition of a graph with an interval colorable graph having a special type of interval coloring also belongs to the class of interval colorable graphs, generalizing earlier findings.

Findings

Regular graphs, complete bipartite graphs, and trees have the special interval coloring.

If G is interval colorable and H has the special interval coloring, then G[H] is interval colorable.

The result applies to trees, extending previous work on regular graphs.

Abstract

An edge-coloring of a graph $G$ with consecutive integers $c_{1},\ldots,c_{t}$ is called an \emph{interval $t$-coloring} if all colors are used, 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 it has an interval $t$-coloring for some positive integer $t$. The set of all interval colorable graphs is denoted by $\mathfrak{N}$. In 2004, Giaro and Kubale showed that if $G,H\in \mathfrak{N}$, then the Cartesian product of these graphs belongs to $\mathfrak{N}$. In the same year they formulated a similar problem for the composition of graphs as an open problem. Later, in 2009, the first author showed that if $G,H\in \mathfrak{N}$ and $H$ is a regular graph, then $G[H]\in \mathfrak{N}$. In this paper, we prove that if $G\in \mathfrak{N}$ and $H$ has an interval coloring of a special type, then $G[H]\in \mathfrak{N}$. Moreover, we show that all regular graphs, complete bipartite graphs and trees have such a special interval coloring. In particular, this implies that if $G\in \mathfrak{N}$ and $T$ is a tree, then $G[T]\in \mathfrak{N}$.