1 research outputs found
Complexity of Computing the Anti-Ramsey Numbers for Paths
The anti-Ramsey numbers are a fundamental notion in graph theory, introduced
in 1978, by Erd\" os, Simonovits and S\' os. For given graphs and the
\emph{anti-Ramsey number} is defined to be the maximum
number such that there exists an assignment of colors to the edges of
in which every copy of in has at least two edges with the same
color.
There are works on the computational complexity of the problem when is a
star. Along this line of research, we study the complexity of computing the
anti-Ramsey number , where is a path of length .
First, we observe that when , the problem is hard; hence, the
challenging part is the computational complexity of the problem when is a
fixed constant.
We provide a characterization of the problem for paths of constant length.
Our first main contribution is to prove that computing for
every integer is NP-hard. We obtain this by providing several structural
properties of such coloring in graphs. We investigate further and show that
approximating to a factor of is hard
already in -partite graphs, unless P=NP. We also study the exact complexity
of the precolored version and show that there is no subexponential algorithm
for the problem unless ETH fails for any fixed constant .
Given the hardness of approximation and parametrization of the problem, it is
natural to study the problem on restricted graph families. We introduce the
notion of color connected coloring and employing this structural property. We
obtain a linear time algorithm to compute , for every
integer , when the host graph, , is a tree