852 research outputs found
More on rainbow disconnection in graphs
Let be a nontrivial edge-colored connected graph. An edge-cut of
is called a rainbow cut if no two edges of it are colored the same. An
edge-colored graph is rainbow disconnected if for every two vertices
and , there exists a rainbow cut. For a connected graph , the
rainbow disconnection number of , denoted by , is defined as the
smallest number of colors that are needed in order to make rainbow
disconnected. In this paper, we first solve a conjecture that determines the
maximum size of a connected graph of order with for given
integers and with , where is odd, posed by
Chartrand et al. in \cite{CDHHZ}. Secondly, we discuss bounds of the rainbow
disconnection numbers for complete multipartite graphs, critical graphs,
minimal graphs with respect to chromatic index and regular graphs, and give the
rainbow disconnection numbers for several special graphs. Finally, we get the
Nordhaus-Gaddum-type theorem for the rainbow disconnection number of graphs. We
prove that if and are both connected, then and . Furthermore, examples are given to show that the upper bounds are
sharp for , and the lower bounds are sharp when .Comment: 14 page
Strong rainbow disconnection in graphs
Let be a nontrivial edge-colored connected graph. An edge-cut of
is called a {\it rainbow edge-cut} if no two edges of are colored with the
same color. For two distinct vertices and of , if an edge-cut
separates them, then the edge-cut is called a {\it --edge-cut}. An
edge-colored graph is called \emph{strong rainbow disconnected} if for
every two distinct vertices and of , there exists a both rainbow and
minimum --edge-cut ({\it rainbow minimum --edge-cut} for short) in
, separating them, and this edge-coloring is called a {\it strong rainbow
disconnection coloring} (srd-{\it coloring} for short) of . For a connected
graph , the \emph{strong rainbow disconnection number} (srd-{\it number} for
short) of , denoted by , is the smallest number of
colors that are needed in order to make strong rainbow disconnected.
In this paper, we first characterize the graphs with edges such that
for each , respectively, and we also
show that the srd-number of a nontrivial connected graph equals the maximum
srd-number among the blocks of . Secondly, we study the srd-numbers for the
complete -partite graphs, -edge-connected -regular graphs and grid
graphs. Finally, we show that for a connected graph , to compute
is NP-hard. In particular, we show that it is already
NP-complete to decide if for a connected cubic graph.
Moreover, we show that for a given edge-colored (with an unbounded number of
colors) connected graph it is NP-complete to decide whether is strong
rainbow disconnected.Comment: 16 pages, 1 figur
The rainbow vertex-disconnection in graphs
Let be a nontrivial connected and vertex-colored graph. A subset of
the vertex set of is called rainbow if any two vertices in have
distinct colors. The graph is called \emph{rainbow vertex-disconnected} if
for any two vertices and of , there exists a vertex subset of
such that when and are nonadjacent, is rainbow and and
belong to different components of ; whereas when and are adjacent,
or is rainbow and and belong to different components of
. For a connected graph , the \emph{rainbow vertex-disconnection
number} of , denoted by , is the minimum number of colors that are
needed to make rainbow vertex-disconnected.
In this paper, we characterize all graphs of order with rainbow
vertex-disconnection number for , and determine the rainbow
vertex-disconnection numbers of some special graphs. Moreover, we study the
extremal problems on the number of edges of a connected graph with order
and for given integers and with .Comment: 18 page
Bounds for the rainbow disconnection number of graphs
An edge-cut of an edge-colored connected graph is called a rainbow-cut if
no two edges in the edge-cut are colored the same. An edge-colored graph is
rainbow disconnected if for any two distinct vertices and of the graph,
there exists a --rainbow-cut separating them. For a connected graph ,
the rainbow disconnection number of , denoted by rd, is defined as the
smallest number of colors that are needed in order to make rainbow
disconnected.
In this paper, we first give some tight upper bounds for rd, and
moreover, we completely characterize the graphs which meet the upper bound of
the Nordhaus-Gaddum type results obtained early by us. Secondly, we propose a
conjecture that , where
is the upper edge-connectivity, and prove the conjecture for
many classes of graphs, to support it. Finally, we give the relationship
between rd of a graph and the rainbow vertex-disconnection number
rvd of the line graph of .Comment: 15 page
Erd\H{o}s-Gallai-type results for the rainbow disconnection number of graphs
Let be a nontrivial connected and edge-colored graph. An edge-cut of
is called a rainbow cut if no two edges of it are colored with a same
color. An edge-colored graph is called rainbow disconnected if for every
two distinct vertices and of , there exists a rainbow cut
separating them. For a connected graph , the rainbow disconnection number of
, denoted by , is defined as the smallest number of colors that are
needed in order to make rainbow disconnected. In this paper, we will study
the Erd\H{o}s-Gallai-type results for , and completely solve them.Comment: 7 pages. arXiv admin note: text overlap with arXiv:1810.0973
Monochromatic disconnection: Erd\H{o}s-Gallai-type problems and product graphs
For an edge-colored graph , we call an edge-cut of monochromatic
if the edges of are colored with a same color. The graph is called
monochromatically disconnected if any two distinct vertices of are
separated by a monochromatic edge-cut. The monochromatic disconnection number,
denoted by , of a connected graph is the maximum number of colors
that are allowed to make monochromatically disconnected. In this paper, we
solve the Erd\H{o}s-Gallai-type problems for the monochromatic disconnection,
and give the monochromatic disconnection numbers for four graph products, i.e.,
Cartesian, strong, lexicographic, and tensor products.Comment: 21 pages, 4 figures. In this new version we obtain the explicit
expression for the extremal function , which was only an upper bound
for the case in the old versio
Graph colorings under global structural conditions
More than ten years ago in 2008, a new kind of graph coloring appeared in
graph theory, which is the {\it rainbow connection coloring} of graphs, and
then followed by some other new concepts of graph colorings, such as {\it
proper connection coloring, monochromatic connection coloring, and
conflict-free connection coloring} of graphs. In about ten years of our
consistent study, we found that these new concepts of graph colorings are
actually quite different from the classic graph colorings. These {\it colored
connection colorings} of graphs are brand-new colorings and they need to take
care of global structural properties (for example, connectivity) of a graph
under the colorings; while the traditional colorings of graphs are colorings
under which only local structural properties (adjacent vertices or edges) of a
graph are taken care of. Both classic colorings and the new colored connection
colorings can produce the so-called chromatic numbers. We call the colored
connection numbers the {\it global chromatic numbers}, and the classic or
traditional chromatic numbers the {\it local chromatic numbers}. This paper
intends to clarify the difference between the colored connection colorings and
the traditional colorings, and finally to propose the new concepts of global
colorings under which global structural properties of the colored graph are
kept, and the global chromatic numbers.Comment: 14 page
Monochromatic disconnection of graphs
For an edge-colored graph , we call an edge-cut of monochromatic
if the edges of are colored with a same color. The graph is called
monochromatically disconnected if any two distinct vertices of are
separated by a monochromatic edge-cut. For a connected graph , the
monochromatic disconnection number, denoted by , of is the maximum
number of colors that are needed in order to make monochromatically
disconnected. We will show that almost all graphs have monochromatic
disconnection numbers equal to 1. We also obtain the Nordhaus-Gaddum-type
results for .Comment: 16 page
Hardness results for rainbow disconnection of graphs
Let be a nontrivial connected, edge-colored graph. An edge-cut of
is called a rainbow cut if no two edges in are colored with a same color.
An edge-coloring of is a rainbow disconnection coloring if for every two
distinct vertices and of , there exists a rainbow cut in
such that and belong to different components of . For a
connected graph , the {\it rainbow disconnection number} of , denoted by
, is defined as the smallest number of colors such that has a
rainbow disconnection coloring by using this number of colors. In this paper,
we show that for a connected graph , computing is NP-hard. In
particular, it is already NP-complete to decide if for a connected
cubic graph. Moreover, we prove that for a given edge-colored (with an
unbounded number of colors) connected graph it is NP-complete to decide
whether is rainbow disconnected.Comment: 8 pages. In the second version we made some correction for the proof
of our main Lemma 2.
Proper disconnection of graphs
For an edge-colored graph , a set of edges of is called a
\emph{proper cut} if is an edge-cut of and any pair of adjacent edges
in are assigned by different colors. An edge-colored graph is \emph{proper
disconnected} if for each pair of distinct vertices of there exists a
proper edge-cut separating them. For a connected graph , the \emph{proper
disconnection number} of , denoted by , is the minimum number of
colors that are needed in order to make proper disconnected. In this paper,
we first give the exact values of the proper disconnection numbers for some
special families of graphs. Next, we obtain a sharp upper bound of for
a connected graph of order , i.e, . Finally, we show that for given integers
and , the minimum size of a connected graph of order with
is for and for .Comment: 14 page
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