Sigma Partitioning: Complexity and Random Graphs
A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors in $P_i$. The $\textit{ sigma number}$ of a graph $G$, deno...
Gespeichert in:
| Veröffentlicht in: | Discrete mathematics and theoretical computer science Jg. 20 no. 2; H. Graph Theory |
|---|---|
| Hauptverfasser: | , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Discrete Mathematics & Theoretical Computer Science
17.12.2018
|
| Schlagworte: | |
| ISSN: | 1365-8050, 1365-8050 |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Abstract | A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors in $P_i$. The $\textit{ sigma number}$ of a graph $G$, denoted by $\sigma(G)$, is the minimum number $k$ such that $ G $ has a sigma partitioning $P_1, \ldots, P_k$. Also, a $\textit{ lucky labeling}$ of a graph $G$ is a function $ \ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $ ($ x \sim y $ means that $ x $ and $y$ are adjacent). The $\textit{ lucky number}$ of $ G $, denoted by $\eta(G)$, is the minimum number $k $ such that $ G $ has a lucky labeling $ \ell :V(G) \rightarrow \mathbb{N}_k$. It was conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is $ \mathbf{NP} $-complete to decide whether $ \eta(G)=2$ for a given 3-regular graph $G$. In this work, we prove this conjecture. Among other results, we give an upper bound of five for the sigma number of a uniformly random graph. |
|---|---|
| AbstractList | A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors in $P_i$. The $\textit{ sigma number}$ of a graph $G$, denoted by $\sigma(G)$, is the minimum number $k$ such that $ G $ has a sigma partitioning $P_1, \ldots, P_k$. Also, a $\textit{ lucky labeling}$ of a graph $G$ is a function $ \ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $ ($ x \sim y $ means that $ x $ and $y$ are adjacent). The $\textit{ lucky number}$ of $ G $, denoted by $\eta(G)$, is the minimum number $k $ such that $ G $ has a lucky labeling $ \ell :V(G) \rightarrow \mathbb{N}_k$. It was conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is $ \mathbf{NP} $-complete to decide whether $ \eta(G)=2$ for a given 3-regular graph $G$. In this work, we prove this conjecture. Among other results, we give an upper bound of five for the sigma number of a uniformly random graph. |
| Author | Dehghan, Ali Sadeghi, Mohammad-Reza Ahadi, Arash |
| Author_xml | – sequence: 1 givenname: Ali surname: Dehghan fullname: Dehghan, Ali – sequence: 2 givenname: Mohammad-Reza surname: Sadeghi fullname: Sadeghi, Mohammad-Reza – sequence: 3 givenname: Arash surname: Ahadi fullname: Ahadi, Arash |
| BookMark | eNpNkM1OwzAQhC1UJNrClXNewGX9l8TcUIBSqQhEy9mynXVx1cRV0gN9e0JBiMvOzhw-jWZCRm1qkZBrBjMuclHe3D-vqxXlQDll-oyMmcgVLUHB6N9_QSZ9vwVgXMtiTGAVN43NXm13iIeY2thubrMqNfsdfsbDMbNtnb0NJzXZvLP7j_6SnAe76_HqV6fk_fFhXT3R5ct8Ud0tqWdSaepQKC-FAAeWI4JVoZDKQimDL5kevFeFEEXpcwyhlrnG3CP3zAEqp1FMyeKHWye7NfsuNrY7mmSjOQWp25jvzn6HBjiWrgYvEJUUKFzQTFnJrVS1564YWLMflu9S33cY_ngMzGk7c9rO8MEapsUXbiljIw |
| ContentType | Journal Article |
| DBID | AAYXX CITATION DOA |
| DOI | 10.23638/DMTCS-20-2-19 |
| DatabaseName | CrossRef DOAJ Directory of Open Access Journals |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | CrossRef |
| Database_xml | – sequence: 1 dbid: DOA name: DOAJ Directory of Open Access Journals url: https://www.doaj.org/ sourceTypes: Open Website |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics Computer Science |
| EISSN | 1365-8050 |
| ExternalDocumentID | oai_doaj_org_article_02e8bd0c3ee543e3bf915a42a45dc2b7 10_23638_DMTCS_20_2_19 |
| GroupedDBID | -~9 .4S .DC 29G 2WC 5GY 5VS 8FE 8FG AAFWJ AAYXX ABDBF ABJCF ABUWG ACGFO ACIWK ACUHS ADBBV ADQAK AENEX AFFHD AFKRA AFPKN AIAGR ALMA_UNASSIGNED_HOLDINGS AMVHM ARCSS B0M BAIFH BBTPI BCNDV BENPR BFMQW BGLVJ BPHCQ CCPQU CITATION EAP EBS ECS EDO EJD EMK EPL EST ESX GROUPED_DOAJ HCIFZ I-F IAO IBB ICD ITC J9A KQ8 KWQ L6V M7S MK~ ML~ OK1 OVT P2P PHGZM PHGZT PIMPY PQGLB PQQKQ PROAC PTHSS PV9 REM RNS RSU RZL TR2 TUS XSB ~8M |
| ID | FETCH-LOGICAL-c1459-be35c4330b0a2ee0a5f745a084fc8190a5c573378c6effd469e6ce2c1b0e5b9e3 |
| IEDL.DBID | DOA |
| ISSN | 1365-8050 |
| IngestDate | Mon Nov 10 04:33:12 EST 2025 Sat Nov 29 08:06:17 EST 2025 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | Graph Theory |
| Language | English |
| License | https://arxiv.org/licenses/nonexclusive-distrib/1.0 |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c1459-be35c4330b0a2ee0a5f745a084fc8190a5c573378c6effd469e6ce2c1b0e5b9e3 |
| OpenAccessLink | https://doaj.org/article/02e8bd0c3ee543e3bf915a42a45dc2b7 |
| ParticipantIDs | doaj_primary_oai_doaj_org_article_02e8bd0c3ee543e3bf915a42a45dc2b7 crossref_primary_10_23638_DMTCS_20_2_19 |
| PublicationCentury | 2000 |
| PublicationDate | 2018-12-17 |
| PublicationDateYYYYMMDD | 2018-12-17 |
| PublicationDate_xml | – month: 12 year: 2018 text: 2018-12-17 day: 17 |
| PublicationDecade | 2010 |
| PublicationTitle | Discrete mathematics and theoretical computer science |
| PublicationYear | 2018 |
| Publisher | Discrete Mathematics & Theoretical Computer Science |
| Publisher_xml | – name: Discrete Mathematics & Theoretical Computer Science |
| SSID | ssj0012947 |
| Score | 2.0935106 |
| Snippet | A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and... |
| SourceID | doaj crossref |
| SourceType | Open Website Index Database |
| SubjectTerms | computer science - computational complexity mathematics - combinatorics |
| Title | Sigma Partitioning: Complexity and Random Graphs |
| URI | https://doaj.org/article/02e8bd0c3ee543e3bf915a42a45dc2b7 |
| Volume | 20 no. 2 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVAON databaseName: DOAJ Directory of Open Access Journals customDbUrl: eissn: 1365-8050 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0012947 issn: 1365-8050 databaseCode: DOA dateStart: 19970101 isFulltext: true titleUrlDefault: https://www.doaj.org/ providerName: Directory of Open Access Journals – providerCode: PRVPQU databaseName: Continental Europe Database customDbUrl: eissn: 1365-8050 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0012947 issn: 1365-8050 databaseCode: BFMQW dateStart: 19970101 isFulltext: true titleUrlDefault: https://search.proquest.com/conteurope providerName: ProQuest – providerCode: PRVPQU databaseName: Engineering Database customDbUrl: eissn: 1365-8050 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0012947 issn: 1365-8050 databaseCode: M7S dateStart: 19970101 isFulltext: true titleUrlDefault: http://search.proquest.com providerName: ProQuest – providerCode: PRVPQU databaseName: ProQuest - Publicly Available Content Database customDbUrl: eissn: 1365-8050 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0012947 issn: 1365-8050 databaseCode: PIMPY dateStart: 19970101 isFulltext: true titleUrlDefault: http://search.proquest.com/publiccontent providerName: ProQuest – providerCode: PRVPQU databaseName: ProQuest Central customDbUrl: eissn: 1365-8050 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0012947 issn: 1365-8050 databaseCode: BENPR dateStart: 19970101 isFulltext: true titleUrlDefault: https://www.proquest.com/central providerName: ProQuest |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrZ07T8MwEMctVBhg4FFAlEflAYkpqp-NzQaFAkOriBapTJHtXFCHtqgtSHx77CSturGwRErkONb_HN2d4_wOoetMaTCK8giUZZH31-3IuLDaZLXl2lCZW1MUm4j7fTUa6WSj1FfYE1bigUvhWoT5XjLiOIAUHLjNNZVGMCNk5pgt_iP3Uc8qmaq-HzAt4hLR6J_OVeuhN-wM_IyIWBSYOhsuaIPUX7iU7iHar2JBfFeO4QhtwbSODlZ1FnD12tXRXm_NVl0cIzIYf0wMTsLQq9XUWxxuCmjL5Q820wy_-sNsgp8CjXpxgt66j8POc1TVPYgcFV4nC1w6wTmxxDAAYmQeC2mIErkLDtxIFyiGsXJtyPPMJ7gQ6no5aglIq4Gfotp0NoUzhImz0hHrm4pMMGosB5mTjOfUxcDa0EA3KynSzxJvkfq0oBAtLURLmT9NqW6g-6DUulXAUhcXvLHSyljpX8Y6_49OLtCuj1pU2FNC40tUW86_4ArtuO_leDFvFvOgibaTl17y_gs1q7p4 |
| linkProvider | Directory of Open Access Journals |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Sigma+Partitioning%3A+Complexity+and+Random+Graphs&rft.jtitle=Discrete+mathematics+and+theoretical+computer+science&rft.au=Ali+Dehghan&rft.au=Mohammad-Reza+Sadeghi&rft.au=Arash+Ahadi&rft.date=2018-12-17&rft.pub=Discrete+Mathematics+%26+Theoretical+Computer+Science&rft.eissn=1365-8050&rft.volume=20+no.+2&rft.issue=Graph+Theory&rft_id=info:doi/10.23638%2FDMTCS-20-2-19&rft.externalDBID=DOA&rft.externalDocID=oai_doaj_org_article_02e8bd0c3ee543e3bf915a42a45dc2b7 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1365-8050&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1365-8050&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1365-8050&client=summon |