1 research outputs found
Interval edge-colorings of composition of graphs
An edge-coloring of a graph with consecutive integers
is called an \emph{interval -coloring} if all colors
are used, and the colors of edges incident to any vertex of are distinct
and form an interval of integers. A graph is interval colorable if it has
an interval -coloring for some positive integer . The set of all interval
colorable graphs is denoted by . In 2004, Giaro and Kubale showed
that if , then the Cartesian product of these graphs
belongs to . 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 and is a regular graph, then
. In this paper, we prove that if and
has an interval coloring of a special type, then .
Moreover, we show that all regular graphs, complete bipartite graphs and trees
have such a special interval coloring. In particular, this implies that if
and is a tree, then .Comment: 12 pages, 3 figure