31,831 research outputs found
Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs
A path in an edge-colored graph is rainbow if no two edges of it are
colored the same. The graph is rainbow-connected if there is a rainbow path
between every pair of vertices. If there is a rainbow shortest path between
every pair of vertices, the graph is strongly rainbow-connected. The
minimum number of colors needed to make rainbow-connected is known as the
rainbow connection number of , and is denoted by . Similarly,
the minimum number of colors needed to make strongly rainbow-connected is
known as the strong rainbow connection number of , and is denoted by
. We prove that for every , deciding whether
is NP-complete for split graphs, which form a subclass
of chordal graphs. Furthermore, there exists no polynomial-time algorithm for
approximating the strong rainbow connection number of an -vertex split graph
with a factor of for any unless P = NP. We
then turn our attention to block graphs, which also form a subclass of chordal
graphs. We determine the strong rainbow connection number of block graphs, and
show it can be computed in linear time. Finally, we provide a polynomial-time
characterization of bridgeless block graphs with rainbow connection number at
most 4.Comment: 13 pages, 3 figure
Conflict-free connection numbers of line graphs
A path in an edge-colored graph is called \emph{conflict-free} if it contains
at least one color used on exactly one of its edges. An edge-colored graph
is \emph{conflict-free connected} if for any two distinct vertices of ,
there is a conflict-free path connecting them. For a connected graph , the
\emph{conflict-free connection number} of , denoted by , is defined
as the minimum number of colors that are required to make conflict-free
connected. In this paper, we investigate the conflict-free connection numbers
of connected claw-free graphs, especially line graphs. We first show that for
an arbitrary connected graph , there exists a positive integer such that
. Secondly, we get the exact value of the conflict-free
connection number of a connected claw-free graph, especially a connected line
graph. Thirdly, we prove that for an arbitrary connected graph and an
arbitrary positive integer , we always have , with only the exception that is isomorphic to a star of order
at least~ and . Finally, we obtain the exact values of ,
and use them as an efficient tool to get the smallest nonnegative integer
such that .Comment: 11 page
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