9,291 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
Rainbow Connection Number and Radius
The rainbow connection number, rc(G), of a connected graph G is the minimum
number of colours needed to colour its edges, so that every pair of its
vertices is connected by at least one path in which no two edges are coloured
the same. In this note we show that for every bridgeless graph G with radius r,
rc(G) <= r(r + 2). We demonstrate that this bound is the best possible for
rc(G) as a function of r, not just for bridgeless graphs, but also for graphs
of any stronger connectivity. It may be noted that for a general 1-connected
graph G, rc(G) can be arbitrarily larger than its radius (Star graph for
instance). We further show that for every bridgeless graph G with radius r and
chordality (size of a largest induced cycle) k, rc(G) <= rk.
It is known that computing rc(G) is NP-Hard [Chakraborty et al., 2009]. Here,
we present a (r+3)-factor approximation algorithm which runs in O(nm) time and
a (d+3)-factor approximation algorithm which runs in O(dm) time to rainbow
colour any connected graph G on n vertices, with m edges, diameter d and radius
r.Comment: Revised preprint with an extra section on an approximation algorithm.
arXiv admin note: text overlap with arXiv:1101.574
Lectures on the topological recursion for Higgs bundles and quantum curves
© 2018 World Scientific Publishing Co. Pte. Ltd. This chapter aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the topological recursion, that has its origin in random matrix theory and has been effectively applied to many enumerative geometry problems; and the other is the quantization of Hitchin spectral curves associated with Higgs bundles. Our emphasis is on explaining the motivation and examples. Concrete examples of the direct relation between Hitchin spectral curves and enumeration problems are given. A general geometric framework of quantum curves is also discussed
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