57 research outputs found
On Interval Non-Edge-Colorable Eulerian Multigraphs
An edge-coloring of a multigraph with colors is called an
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. In this
note, we show that all Eulerian multigraphs with an odd number of edges have no
interval coloring. We also give some methods for constructing of interval
non-edge-colorable Eulerian multigraphs.Comment: 4 page
Interval Edge Colourings of Complete Graphs and n-cubes
For complete graphs and n-cubes bounds are found for the possible number of
colours in an interval edge colourings.Comment: 4 page
On interval edge-colorings of outerplanar graphs
An edge-coloring of a graph with colors is called an
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 . For an interval colorable graph , the least value of for
which has an interval -coloring is denoted by . A graph is
outerplanar if it can be embedded in the plane so that all its vertices lie on
the same (unbounded) face. In this paper we show that if is a 2-connected
outerplanar graph with , then is interval colorable and
\begin{center} w(G)=\left\{\begin{tabular}{ll} 3, & if | V(G)| is even, \ 4,
& if | V(G)| is odd. \end{tabular}% \right. \end{center} We also give a
negative answer to the question of Axenovich on the outerplanar triangulations.Comment: 9 pages, 3 figure
Interval edge colorings of some products of graphs
An edge coloring of a graph with colors is called an
interval -coloring if for each there is at least one
edge of colored by , and the colors of edges incident to any vertex of
are distinct and form an interval of integers. A graph is interval
colorable, if there is an integer for which has an interval
-coloring. Let be the set of all interval colorable graphs.
In 2004 Kubale and Giaro showed that if , then the
Cartesian product of these graphs belongs to . Also, they
formulated a similar problem for the lexicographic product as an open problem.
In this paper we first show that if , then for any . Furthermore, we show that if and is a regular graph, then strong and lexicographic
products of graphs belong to . We also prove that tensor
and strong tensor products of graphs belong to if and is a regular graph.Comment: 14 pages, 5 figures, minor change
Interval colorings of complete balanced multipartite graphs
A graph is called a complete -partite () graph if its
vertices can be partitioned into independent sets such
that each vertex in is adjacent to all the other vertices in
for . A complete -partite graph is a complete balanced
-partite graph if . 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 . In this paper we show that
a complete balanced -partite graph with vertices in each part is
interval colorable if and only if is even. We also prove that if is
even and , then a complete balanced -partite
graph admits an interval -coloring. Moreover, if , where
is odd and , then a complete balanced -partite graph
has an interval -coloring for each positive integer satisfying
.Comment: 10 page
Sequential edge-coloring on the subset of vertices of almost regular graphs
Let be a graph and . A proper edge-coloring of a graph
with colors is called an -sequential -coloring if the
edges incident to each vertex are colored by the colors
, where is the degree of the vertex in .
In this note, we show that if is a graph with
and (), then has an -sequential
-coloring with , where and
. As a corollary, we obtain the
following result: if is a graph with and
(), then , where
is the edge-chromatic sum of .Comment: 4 page
On maximum matchings in almost regular graphs
In 2010, Mkrtchyan, Petrosyan and Vardanyan proved that every graph with
contains a maximum matching whose
unsaturated vertices do not have a common neighbor, where and
denote the maximum and minimum degrees of vertices in ,
respectively. In the same paper they suggested the following conjecture: every
graph with contains a maximum matching whose
unsaturated vertices do not have a common neighbor. Recently, Picouleau
disproved this conjecture by constructing a bipartite counterexample with
and . In this note we show that the conjecture is
false for graphs with and , and
for -regular graphs when .Comment: 5 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 Total Colorings of Complete Multipartite Graphs and Hypercubes
A total coloring of a graph is a coloring of its vertices and edges such
that no adjacent vertices, edges, and no incident vertices and edges obtain the
same color. An interval total -coloring of a graph is a total coloring
of with colors such that all colors are used, and the edges
incident to each vertex together with are colored by
consecutive colors, where is the degree of a vertex in . In
this paper we prove that all complete multipartite graphs with the same number
of vertices in each part are interval total colorable. Moreover, we also give
some bounds for the minimum and the maximum span in interval total colorings of
these graphs. Next, we investigate interval total colorings of hypercubes
. In particular, we prove that () has an interval total
-coloring if and only if .Comment: 17 page
On Lower Bound for W(K_{2n})
The lower bound W(K_{2n})>=3n-2 is proved for the greatest possible number of
colors in an interval edge coloring of the complete graph K_{2n}.Comment: 3 page
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