Bijections in de Bruijn Graphs

Kód výsledku v 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>

Výsledek na webu

<a href="http://www.combinatorialmath.ca/ArsCombinatoria/Vol143.html" target="_blank" >http://www.combinatorialmath.ca/ArsCombinatoria/Vol143.html</a>

DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

Bijections in de Bruijn Graphs

Popis výsledku v původním jazyce

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 $S(2^{n-1})$.

Název v anglickém jazyce

Bijections in de Bruijn Graphs

Popis výsledku anglicky

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 $S(2^{n-1})$.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

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Návaznosti

S - Specificky vyzkum na vysokych skolach

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Ars Combinatoria

ISSN

0381-7032

e-ISSN

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Svazek periodika

2019

Číslo periodika v rámci svazku

143

Stát vydavatele periodika

CA - Kanada

Počet stran výsledku

12

Strana od-do

215-226

Kód UT WoS článku

000476581000018

EID výsledku v databázi Scopus

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