Bijections in de Bruijn Graphs

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F19%3A00335007" target="_blank" >RIV/68407700:21340/19:00335007 - isvavai.cz</a>
Result on the web
<a href="http://www.combinatorialmath.ca/ArsCombinatoria/Vol143.html" target="_blank" >http://www.combinatorialmath.ca/ArsCombinatoria/Vol143.html</a>
DOI - Digital Object Identifier
—

Result language
angličtina
Original language name
Bijections in de Bruijn Graphs
Original language description
A T-net of order $m$ is a graph with $m$ nodes and $2m$ directed edges, where every node has indegree and outdegree equal to $2$. (A well known example of T-nets are de Bruijn graphs.) Given a T-net $N$ of order $m$, there is the so called "doubling" process that creates a T-net $N^*$ from $N$ with $2m$ nodes and $4m$ edges. Let $vert Xvert$ denote the number of Eulerian cycles in a graph $X$. It is known that $vert N^*vert=2^{m-1}vert Nvert$. In this paper we present a new proof of this identity. Moreover we prove that $vert Nvertleq 2^{m-1}$. Let $Theta(X)$ denote the set of all Eulerian cycles in a graph $X$ and $S(n)$ the set of all binary sequences of length $n$. Exploiting the new proof we construct a bijection $Theta(N)times S(m-1)rightarrow Theta(N^*)$, which allows us to solve one of Stanley's open questions: we find a bijection between de Bruijn sequences of order $n$ and $S(2^{n-1})$.
Czech name
—
Czech description
—

Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Similar results(10)