Star transposition Gray codes for multiset permutations

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F23%3A10476030" target="_blank" >RIV/00216208:11320/23:10476030 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=0k8Yngo_6b" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=0k8Yngo_6b</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1002/jgt.22915" target="_blank" >10.1002/jgt.22915</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Star transposition Gray codes for multiset permutations

Popis výsledku v původním jazyce

Given integers k &gt;= 2 and a_1,...,a_k &gt;= 1, let a:= (a_1,...,a_k) and n:= a_1+...+a_k. An a-multiset permutation is a string of length n that contains exactly a_i symbols i for each i=1,...,k. In this work we consider the problem of exhaustively generating all a-multiset permutations by star transpositions, that is, in each step, the first entry of the string is transposed with any other entry distinct from the first one. This is a far-ranging generalization of several known results. For example, it is known that permutations (a_1=...=a_k=1) can be generated by star transpositions, while combinations (k=2) can be generated by these operations if and only if they are balanced (a_1=a_2), with the positive case following from the middle levels theorem. To understand the problem in general, we introduce a parameter Delta(a):= n-2max{a_1,...,a_k} that allows us to distinguish three different regimes for this problem. We show that if Delta(a)&lt;0, then a star transposition Gray code for a-multiset permutations does not exist. We also construct such Gray codes for the case Delta(a)&gt;0, assuming that they exist for the case Delta(a)=0. For the case Delta(a)=0 we present some partial positive results. Our proofs establish Hamilton-connectedness or Hamilton-laceability of the underlying flip graphs, and they answer several cases of a recent conjecture of Shen and Williams. In particular, we prove that the middle levels graph is Hamilton-laceable.

Název v anglickém jazyce

Star transposition Gray codes for multiset permutations

Popis výsledku anglicky

Given integers k &gt;= 2 and a_1,...,a_k &gt;= 1, let a:= (a_1,...,a_k) and n:= a_1+...+a_k. An a-multiset permutation is a string of length n that contains exactly a_i symbols i for each i=1,...,k. In this work we consider the problem of exhaustively generating all a-multiset permutations by star transpositions, that is, in each step, the first entry of the string is transposed with any other entry distinct from the first one. This is a far-ranging generalization of several known results. For example, it is known that permutations (a_1=...=a_k=1) can be generated by star transpositions, while combinations (k=2) can be generated by these operations if and only if they are balanced (a_1=a_2), with the positive case following from the middle levels theorem. To understand the problem in general, we introduce a parameter Delta(a):= n-2max{a_1,...,a_k} that allows us to distinguish three different regimes for this problem. We show that if Delta(a)&lt;0, then a star transposition Gray code for a-multiset permutations does not exist. We also construct such Gray codes for the case Delta(a)&gt;0, assuming that they exist for the case Delta(a)=0. For the case Delta(a)=0 we present some partial positive results. Our proofs establish Hamilton-connectedness or Hamilton-laceability of the underlying flip graphs, and they answer several cases of a recent conjecture of Shen and Williams. In particular, we prove that the middle levels graph is Hamilton-laceable.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/GA19-08554S" target="_blank" >GA19-08554S: Struktury a algoritmy ve velmi symetrických grafech</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2023

Kód důvěrnosti údajů

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

Název periodika

Journal of Graph Theory

ISSN

0364-9024

e-ISSN

1097-0118

Svazek periodika

103

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

59

Strana od-do

212-270

Kód UT WoS článku

000909562200001

EID výsledku v databázi Scopus

2-s2.0-85146162944