47 research outputs found
Bijective mapping preserving intersecting antichains for k-valued cubes
Generalizing a result of Miyakawa, Nozaki, Pogosyan and Rosenberg, we prove
that there is a one-to-one correspondence between the set of intersecting
antichains in a subset of the lower half of the k-valued n-cube and the set of
intersecting antichains in the k-valued (n-1)-cube.Comment: 6 page
On covering expander graphs by Hamilton cycles
The problem of packing Hamilton cycles in random and pseudorandom graphs has
been studied extensively. In this paper, we look at the dual question of
covering all edges of a graph by Hamilton cycles and prove that if a graph with
maximum degree satisfies some basic expansion properties and contains
a family of edge disjoint Hamilton cycles, then there also
exists a covering of its edges by Hamilton cycles. This
implies that for every and every there exists
a covering of all edges of by Hamilton cycles
asymptotically almost surely, which is nearly optimal.Comment: 19 pages. arXiv admin note: some text overlap with arXiv:some
math/061275
Conflict-free coloring of graphs
We study the conflict-free chromatic number chi_{CF} of graphs from extremal
and probabilistic point of view. We resolve a question of Pach and Tardos about
the maximum conflict-free chromatic number an n-vertex graph can have. Our
construction is randomized. In relation to this we study the evolution of the
conflict-free chromatic number of the Erd\H{o}s-R\'enyi random graph G(n,p) and
give the asymptotics for p=omega(1/n). We also show that for p \geq 1/2 the
conflict-free chromatic number differs from the domination number by at most 3.Comment: 12 page
Compactness and finite forcibility of graphons
Graphons are analytic objects associated with convergent sequences of graphs.
Problems from extremal combinatorics and theoretical computer science led to a
study of graphons determined by finitely many subgraph densities, which are
referred to as finitely forcible. Following the intuition that such graphons
should have finitary structure, Lovasz and Szegedy conjectured that the
topological space of typical vertices of a finitely forcible graphon is always
compact. We disprove the conjecture by constructing a finitely forcible graphon
such that the associated space is not compact. The construction method gives a
general framework for constructing finitely forcible graphons with non-trivial
properties
The number of Hamiltonian decompositions of regular graphs
A Hamilton cycle in a graph is a cycle passing through every vertex
of . A Hamiltonian decomposition of is a partition of its edge
set into disjoint Hamilton cycles. One of the oldest results in graph theory is
Walecki's theorem from the 19th century, showing that a complete graph on
an odd number of vertices has a Hamiltonian decomposition. This result was
recently greatly extended by K\"{u}hn and Osthus. They proved that every
-regular -vertex graph with even degree for some fixed
has a Hamiltonian decomposition, provided is sufficiently
large. In this paper we address the natural question of estimating ,
the number of such decompositions of . Our main result is that
. In particular, the number of Hamiltonian
decompositions of is
Extremal graphs for clique-paths
In this paper we deal with a Tur\'an-type problem: given a positive integer n
and a forbidden graph H, how many edges can there be in a graph on n vertices
without a subgraph H? How does a graph look like if it has this extremal edge
number?
The forbidden graph in this article is a clique-path: a path of length k
where each edge is extended to an r-clique, r >2. We determine both the
extremal number and the extremal graphs for sufficiently large n.Comment: 12 pages, 7 figure