43 research outputs found
On the path-avoidance vertex-coloring game
For any graph and any integer , the \emph{online vertex-Ramsey
density of and }, denoted , is a parameter defined via a
deterministic two-player Ramsey-type game (Painter vs.\ Builder). This
parameter was introduced in a recent paper \cite{mrs11}, where it was shown
that the online vertex-Ramsey density determines the threshold of a similar
probabilistic one-player game (Painter vs.\ the binomial random graph
). For a large class of graphs , including cliques, cycles,
complete bipartite graphs, hypercubes, wheels, and stars of arbitrary size, a
simple greedy strategy is optimal for Painter and closed formulas for
are known. In this work we show that for the case where
is a (long) path, the picture is very different. It is not hard to see that
for an appropriately defined integer
, and that the greedy strategy gives a lower bound of
. We construct and analyze Painter strategies that
improve on this greedy lower bound by a factor polynomial in , and we
show that no superpolynomial improvement is possible
A constant-time algorithm for middle levels Gray codes
For any integer a middle levels Gray code is a cyclic listing of
all -element and -element subsets of such that
any two consecutive subsets differ in adding or removing a single element. The
question whether such a Gray code exists for any has been the subject
of intensive research during the last 30 years, and has been answered
affirmatively only recently [T. M\"utze. Proof of the middle levels conjecture.
Proc. London Math. Soc., 112(4):677--713, 2016]. In a follow-up paper [T.
M\"utze and J. Nummenpalo. An efficient algorithm for computing a middle levels
Gray code. To appear in ACM Transactions on Algorithms, 2018] this existence
proof was turned into an algorithm that computes each new set in the Gray code
in time on average. In this work we present an algorithm for
computing a middle levels Gray code in optimal time and space: each new set is
generated in time on average, and the required space is
Efficient computation of middle levels Gray codes
For any integer a middle levels Gray code is a cyclic listing of
all bitstrings of length that have either or entries equal to
1 such that any two consecutive bitstrings in the list differ in exactly one
bit. The question whether such a Gray code exists for every has been
the subject of intensive research during the last 30 years, and has been
answered affirmatively only recently [T. M\"utze. Proof of the middle levels
conjecture. Proc. London Math. Soc., 112(4):677--713, 2016]. In this work we
provide the first efficient algorithm to compute a middle levels Gray code. For
a given bitstring, our algorithm computes the next bitstrings in the
Gray code in time , which is
on average per bitstring provided that
On Hamilton cycles in graphs defined by intersecting set systems
In 1970 Lov\'asz conjectured that every connected vertex-transitive graph
admits a Hamilton cycle, apart from five exceptional graphs. This conjecture
has recently been settled for graphs defined by intersecting set systems, which
feature prominently throughout combinatorics. In this expository article, we
retrace these developments and give an overview of the many different
ingredients in the proofs
A book proof of the middle levels theorem
We give a short constructive proof for the existence of a Hamilton cycle in
the subgraph of the -dimensional hypercube induced by all vertices with
exactly or many 1s
A short proof of the middle levels theorem
Consider the graph that has as vertices all bitstrings of length with
exactly or entries equal to 1, and an edge between any two bitstrings
that differ in exactly one bit. The well-known middle levels conjecture asserts
that this graph has a Hamilton cycle for any . In this paper we
present a new proof of this conjecture, which is much shorter and more
accessible than the original proof
Sparse Kneser graphs are Hamiltonian
For integers and , the Kneser graph is the
graph whose vertices are the -element subsets of and whose
edges connect pairs of subsets that are disjoint. The Kneser graphs of the form
are also known as the odd graphs. We settle an old problem due to
Meredith, Lloyd, and Biggs from the 1970s, proving that for every ,
the odd graph has a Hamilton cycle. This and a known conditional
result due to Johnson imply that all Kneser graphs of the form
with and have a Hamilton cycle. We also prove that
has at least distinct Hamilton cycles for .
Our proofs are based on a reduction of the Hamiltonicity problem in the odd
graph to the problem of finding a spanning tree in a suitably defined
hypergraph on Dyck words