915 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
The anti-Ramsey threshold of complete graphs
For graphs and , let G {\displaystyle\smash{\begin{subarray}{c}
\hbox{\tiny\rm rb} \\ \longrightarrow \\ \hbox{\tiny\rm p}
\end{subarray}}}H denote the property that for every proper edge-colouring of
there is a rainbow in . It is known that, for every graph , an
asymptotic upper bound for the threshold function of this property for the random graph is
, where denotes the so-called maximum
-density of . Extending a result of Nenadov, Person, \v{S}kori\'c, and
Steger [J. Combin. Theory Ser. B 124 (2017),1-38] we prove a matching lower
bound for for . Furthermore, we show that .Comment: 19 page
Extremal properties of flood-filling games
The problem of determining the number of "flooding operations" required to
make a given coloured graph monochromatic in the one-player combinatorial game
Flood-It has been studied extensively from an algorithmic point of view, but
basic questions about the maximum number of moves that might be required in the
worst case remain unanswered. We begin a systematic investigation of such
questions, with the goal of determining, for a given graph, the maximum number
of moves that may be required, taken over all possible colourings. We give
several upper and lower bounds on this quantity for arbitrary graphs and show
that all of the bounds are tight for trees; we also investigate how much the
upper bounds can be improved if we restrict our attention to graphs with higher
edge-density.Comment: Final version, accepted to DMTC
- β¦