79,270 research outputs found
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
Poset-free Families and Lubell-boundedness
Given a finite poset , we consider the largest size \lanp of a family
\F of subsets of that contains no subposet . This
continues the study of the asymptotic growth of \lanp; it has been
conjectured that for all , \pi(P):= \lim_{n\rightarrow\infty} \lanp/\nchn
exists and equals a certain integer, . While this is known to be true for
paths, and several more general families of posets, for the simple diamond
poset \D_2, the existence of frustratingly remains open. Here we
develop theory to show that exists and equals the conjectured value
for many new posets . We introduce a hierarchy of properties for
posets, each of which implies , and some implying more precise
information about \lanp. The properties relate to the Lubell function of a
family \F of subsets, which is the average number of times a random full
chain meets \F. We present an array of examples and constructions that
possess the properties
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
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