Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Interval edge-colorings of composition of graphs   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 $G[T]\in \mathfrak{N}$.

6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an
20
The categories of this paper are: acc-phys
9
The categories of this paper are: nlin
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th
19
The categories of this paper are: eess
0
The categories of this paper are: hep-ph