75 research outputs found
An explicit universal cycle for the (n-1)-permutations of an n-set
We show how to construct an explicit Hamilton cycle in the directed Cayley
graph Cay({\sigma_n, sigma_{n-1}} : \mathbb{S}_n), where \sigma_k = (1 2 >...
k). The existence of such cycles was shown by Jackson (Discrete Mathematics,
149 (1996) 123-129) but the proof only shows that a certain directed graph is
Eulerian, and Knuth (Volume 4 Fascicle 2, Generating All Tuples and
Permutations (2005)) asks for an explicit construction. We show that a simple
recursion describes our Hamilton cycle and that the cycle can be generated by
an iterative algorithm that uses O(n) space. Moreover, the algorithm produces
each successive edge of the cycle in constant time; such algorithms are said to
be loopless
Subset-lex: did we miss an order?
We generalize a well-known algorithm for the generation of all subsets of a
set in lexicographic order with respect to the sets as lists of elements
(subset-lex order). We obtain algorithms for various combinatorial objects such
as the subsets of a multiset, compositions and partitions represented as lists
of parts, and for certain restricted growth strings. The algorithms are often
loopless and require at most one extra variable for the computation of the next
object. The performance of the algorithms is very competitive even when not
loopless. A Gray code corresponding to the subset-lex order and a Gray code for
compositions that was found during this work are described.Comment: Two obvious errors corrected (indicated by "Correction:" in the LaTeX
source
Lattice path matroids: enumerative aspects and Tutte polynomials
Fix two lattice paths P and Q from (0,0) to (m,r) that use East and North
steps with P never going above Q. We show that the lattice paths that go from
(0,0) to (m,r) and that remain in the region bounded by P and Q can be
identified with the bases of a particular type of transversal matroid, which we
call a lattice path matroid. We consider a variety of enumerative aspects of
these matroids and we study three important matroid invariants, namely the
Tutte polynomial and, for special types of lattice path matroids, the
characteristic polynomial and the beta invariant. In particular, we show that
the Tutte polynomial is the generating function for two basic lattice path
statistics and we show that certain sequences of lattice path matroids give
rise to sequences of Tutte polynomials for which there are relatively simple
generating functions. We show that Tutte polynomials of lattice path matroids
can be computed in polynomial time. Also, we obtain a new result about lattice
paths from an analysis of the beta invariant of certain lattice path matroids.Comment: 28 pages, 11 figure
Transitive and Gallai colorings
A Gallai coloring of the complete graph is an edge-coloring with no rainbow
triangle. This concept first appeared in the study of comparability graphs and
anti-Ramsey theory. We introduce a transitive analogue for acyclic directed
graphs, and generalize both notions to Coxeter systems, matroids and
commutative algebras.
It is shown that for any finite matroid (or oriented matroid), the maximal
number of colors is equal to the matroid rank. This generalizes a result of
Erd\H{o}s-Simonovits-S\'os for complete graphs. The number of Gallai (or
transitive) colorings of the matroid that use at most colors is a
polynomial in . Also, for any acyclic oriented matroid, represented over the
real numbers, the number of transitive colorings using at most 2 colors is
equal to the number of chambers in the dual hyperplane arrangement.
We count Gallai and transitive colorings of the root system of type A using
the maximal number of colors, and show that, when equipped with a natural
descent set map, the resulting quasisymmetric function is symmetric and
Schur-positive.Comment: 31 pages, 5 figure
On some third parts of nearly complete digraphs
AbstractFor the complete digraph DKn with n⩾3, its half as well as its third (or near-third) part, both non-self-converse, are exhibited. A backtracking method for generating a tth part of a digraph is sketched. It is proved that some self-converse digraphs are not among the near-third parts of the complete digraph DK5 in four of the six possible cases. For n=9+6k,k=0,1,…, a third part D of DKn is found such that D is a self-converse oriented graph and all D-decompositions of DKn have trivial automorphism group
Ramanujan Coverings of Graphs
Let be a finite connected graph, and let be the spectral radius of
its universal cover. For example, if is -regular then
. We show that for every , there is an -covering
(a.k.a. an -lift) of where all the new eigenvalues are bounded from
above by . It follows that a bipartite Ramanujan graph has a Ramanujan
-covering for every . This generalizes the case due to Marcus,
Spielman and Srivastava (2013).
Every -covering of corresponds to a labeling of the edges of by
elements of the symmetric group . We generalize this notion to labeling
the edges by elements of various groups and present a broader scenario where
Ramanujan coverings are guaranteed to exist.
In particular, this shows the existence of richer families of bipartite
Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava,
a crucial component of our proof is the existence of interlacing families of
polynomials for complex reflection groups. The core argument of this component
is taken from a recent paper of them (2015).
Another important ingredient of our proof is a new generalization of the
matching polynomial of a graph. We define the -th matching polynomial of
to be the average matching polynomial of all -coverings of . We show this
polynomial shares many properties with the original matching polynomial. For
example, it is real rooted with all its roots inside .Comment: 38 pages, 4 figures, journal version (minor changes from previous
arXiv version). Shortened version appeared in STOC 201
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