14 research outputs found
Rainbow Hamiltonicity and the spectral radius
Let be a family of graphs of order with
the same vertex set. A rainbow Hamiltonian cycle in is a cycle
that visits each vertex precisely once such that any two edges belong to
different graphs of . We obtain a rainbow version of Ore's size
condition of Hamiltonicity, and pose a related problem. Towards a solution of
that problem, we give a sufficient condition for the existence of a rainbow
Hamiltonian cycle in terms of the spectral radii of the graphs in
and completely characterize the corresponding extremal graphs
Bounds for the collapsibility number of a simplicial complex and non-cover complexes of hypergraphs
The collapsibility number of simplicial complexes was introduced by Wegner in
order to understand the intersection patterns of convex sets. This number also
plays an important role in a variety of Helly type results. There are only a
few upper bounds for the collapsibility number of complexes available in
literature. In general, it is difficult to establish such non-trivial upper
bounds. In this article, we construct a sequence of upper bounds
for the collapsibility number of a simplicial complex . We also show that
the bound given by is tight if the underlying complex is -vertex
decomposable. We then give an upper bound for and therefore for the
collapsibility number of the non-cover complex of a hypergraph in terms of its
covering number
Cooperative conditions for the existence of rainbow matchings
Let , and let be a family of non-empty sets of
edges in a bipartite graph. If the union of every members of
contains a matching of size , then there exists an -rainbow
matching of size . Upon replacing by , the result can be
proved both topologically and by a relatively simple combinatorial argument.
The main effort is in gaining the last , which makes the result sharp
A system of disjoint representatives of line segments with given directions
We prove that for all positive integers and , there exists an integer
satisfying the following. If is a set of direction vectors
in the plane and is the set of all line segments in direction
for some , then for every families , each consisting of mutually disjoint segments in
, there is a set of disjoint segments
in and distinct integers satisfying that for all
. We generalize this property for underlying lines on
fixed directions to families of simple curves with certain conditions
Badges and rainbow matchings
Drisko proved that matchings of size in a bipartite graph have a
rainbow matching of size . For general graphs it is conjectured that
matchings suffice for this purpose (and that matchings suffice when
is even). The known graphs showing sharpness of this conjecture for even
are called badges. We improve the previously best known bound from to
, using a new line of proof that involves analysis of the appearance of
badges. We also prove a "cooperative" generalization: for and ,
any sets of edges, the union of every of which contains a matching
of size , have a rainbow matching of size .Comment: Accepted for publication in Discrete Mathematics. 19 pages, 2 figure
Optimal Bounds for the Colorful Fractional Helly Theorem
The well known fractional Helly theorem and colorful Helly theorem can be
merged into the so called colorful fractional Helly theorem. It states: For
every and every non-negative integer , there is
with the following property. Let
be finite nonempty families of convex
sets in of sizes respectively. If at least
of the colorful -tuples have a nonempty
intersection, then there is such that contains a
subfamily of size at least with a nonempty intersection. (A
colorful -tuple is a -tuple such that
belongs to for every .)
The colorful fractional Helly theorem was first stated and proved by
B\'ar\'any, Fodor, Montejano, Oliveros, and P\'or in 2014 with . In 2017 Kim proved the theorem with better function
, which in particular tends to when tends to . Kim
also conjectured what is the optimal bound for and
provided the upper bound example for the optimal bound. The conjectured bound
coincides with the optimal bounds for the (non-colorful) fractional Helly
theorem proved independently by Eckhoff and Kalai around 1984.
We verify Kim's conjecture by extending Kalai's approach to the colorful
scenario. Moreover, we obtain optimal bounds also in more general setting when
we allow several sets of the same color.Comment: 13 pages, 1 figure. The main technical result is extended to c
colors, where c is a positive integer, in contrast to the previous version
where we only allowed (d+1) colors. We added the acknowledgment