Kneser Graphs Are Hamiltonian

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F23%3A10476049" target="_blank" >RIV/00216208:11320/23:10476049 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1145/3564246.3585137" target="_blank" >https://doi.org/10.1145/3564246.3585137</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1145/3564246.3585137" target="_blank" >10.1145/3564246.3585137</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Kneser Graphs Are Hamiltonian

Popis výsledku v původním jazyce

For integers k &gt;= 1 and n &gt;= 2k + 1, the Kneser graph K (n, k) has as vertices all k-element subsets of an n-element ground set, and an edge between any two disjoint sets. It has been conjectured since the 1970s that all Kneser graphs admit a Hamilton cycle, with one notable exception, namely the Petersen graph.. (5, 2). This problem received considerable attention in the literature, including a recent solution for the sparsest case n = 2k + 1. The main contribution of this paper is to prove the conjecture in full generality. We also extend this Hamiltonicity result to all connected generalized Johnson graphs (except the Petersen graph). The generalized Johnson graph J (n, k, s) has as vertices all k-element subsets of an n-element ground set, and an edge between any two sets whose intersection has size exactly s. Clearly, we have K (n, k) = J (n, k, 0), i.e., generalized Johnson graphs include Kneser graphs as a special case. Our results imply that all known families of vertex-transitive graphs defined by intersecting set systems have a Hamilton cycle, which settles an interesting special case of Lovasz&apos; conjecture on Hamilton cycles in vertex-transitive graphs from 1970. Our main technical innovation is to study cycles in Kneser graphs by a kinetic system of multiple gliders that move at different speeds and that interact over time, reminiscent of the gliders in Conway&apos;s Game of Life, and to analyze this system combinatorially and via linear algebra.

Název v anglickém jazyce

Kneser Graphs Are Hamiltonian

Popis výsledku anglicky

For integers k &gt;= 1 and n &gt;= 2k + 1, the Kneser graph K (n, k) has as vertices all k-element subsets of an n-element ground set, and an edge between any two disjoint sets. It has been conjectured since the 1970s that all Kneser graphs admit a Hamilton cycle, with one notable exception, namely the Petersen graph.. (5, 2). This problem received considerable attention in the literature, including a recent solution for the sparsest case n = 2k + 1. The main contribution of this paper is to prove the conjecture in full generality. We also extend this Hamiltonicity result to all connected generalized Johnson graphs (except the Petersen graph). The generalized Johnson graph J (n, k, s) has as vertices all k-element subsets of an n-element ground set, and an edge between any two sets whose intersection has size exactly s. Clearly, we have K (n, k) = J (n, k, 0), i.e., generalized Johnson graphs include Kneser graphs as a special case. Our results imply that all known families of vertex-transitive graphs defined by intersecting set systems have a Hamilton cycle, which settles an interesting special case of Lovasz&apos; conjecture on Hamilton cycles in vertex-transitive graphs from 1970. Our main technical innovation is to study cycles in Kneser graphs by a kinetic system of multiple gliders that move at different speeds and that interact over time, reminiscent of the gliders in Conway&apos;s Game of Life, and to analyze this system combinatorially and via linear algebra.

Druh

D - Stať ve sborníku

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/GA22-15272S" target="_blank" >GA22-15272S: Principy kombinatorického generování</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 statě ve sborníku

PROCEEDINGS OF THE 55TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING, STOC 2023

ISBN

978-1-4503-9913-5

ISSN

—

e-ISSN

—

Počet stran výsledku

8

Strana od-do

963-970

Název nakladatele

ASSOC COMPUTING MACHINERY

Místo vydání

NEW YORK

Místo konání akce

Orlando, USA

Datum konání akce

20. 6. 2023

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

001064640700079