2 research outputs found
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