4 research outputs found
Interval edge-colorings of graph products
An interval t-coloring of a graph G is a proper edge-coloring of G with
colors 1,2,...,t such that at least one edge of G is colored by i, i=1,2,...,t,
and the edges incident to each vertex v\in V(G) are colored by d_{G}(v)
consecutive colors, where d_{G}(v) is the degree of the vertex v in G. In this
paper interval edge-colorings of various graph products are investigated.Comment: 4 page
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
Interval edge-colorings of Cartesian products of graphs I
An edge-coloring of a graph with colors is an interval
-coloring if all colors are used, and the colors of edges incident to each
vertex of are distinct and form an interval of integers. A graph is
interval colorable if has an interval -coloring for some positive
integer . Let be the set of all interval colorable graphs.
For a graph , the least and the greatest values of for
which has an interval -coloring are denoted by and ,
respectively. In this paper we first show that if is an -regular graph
and , then
() and ().
Next, we investigate interval edge-colorings of grids, cylinders and tori. In
particular, we prove that if is planar and both factors have at
least 3 vertices, then and .
Finally, we confirm the first author's conjecture on the -dimensional cube
and show that has an interval -coloring if and only if
.Comment: 18 page
Interval cyclic edge-colorings of graphs
A proper edge-coloring of a graph with colors is called an
\emph{interval cyclic -coloring} if all colors are used, and the edges
incident to each vertex are colored by consecutive
colors modulo , where is the degree of a vertex in . A
graph is \emph{interval cyclically colorable} if it has an interval cyclic
-coloring for some positive integer . The set of all interval cyclically
colorable graphs is denoted by . For a graph , the least and the greatest values of for which it has an
interval cyclic -coloring are denoted by and ,
respectively. In this paper we investigate some properties of interval cyclic
colorings. In particular, we prove that if is a triangle-free graph with at
least two vertices and , then . We also obtain bounds on and for
various classes of graphs. Finally, we give some methods for constructing of
interval cyclically non-colorable graphs.Comment: 23 pages, 5 figure