{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T09:58:42Z","timestamp":1772013522843,"version":"3.50.1"},"reference-count":17,"publisher":"EDP Sciences","issue":"1","license":[{"start":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T00:00:00Z","timestamp":1771977600000},"content-version":"vor","delay-in-days":55,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2025,12,20]]},"published-print":{"date-parts":[[2026,1]]},"abstract":"<jats:p>\n                    A partition \u03c0 = {\n                    <jats:italic>V<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    ,\n                    <jats:italic>V<\/jats:italic>\n                    <jats:sub>2<\/jats:sub>\n                    , \u2026,\n                    <jats:italic>\n                      V\n                      <jats:sub>k<\/jats:sub>\n                    <\/jats:italic>\n                    } of the vertex set\n                    <jats:italic>V<\/jats:italic>\n                    of a graph\n                    <jats:italic>G<\/jats:italic>\n                    into\n                    <jats:italic>k<\/jats:italic>\n                    color classes\n                    <jats:italic>\n                      V\n                      <jats:sub>i<\/jats:sub>\n                    <\/jats:italic>\n                    , with 1 \u2264\n                    <jats:italic>i<\/jats:italic>\n                    \u2264\n                    <jats:italic>k<\/jats:italic>\n                    is called a quorum coloring of\n                    <jats:italic>G<\/jats:italic>\n                    if for every vertex\n                    <jats:italic>v<\/jats:italic>\n                    \u2208\n                    <jats:italic>V<\/jats:italic>\n                    , at least half of the vertices in the closed neighborhood\n                    <jats:italic>N<\/jats:italic>\n                    [\n                    <jats:italic>v<\/jats:italic>\n                    ] of\n                    <jats:italic>v<\/jats:italic>\n                    have the same color as\n                    <jats:italic>v<\/jats:italic>\n                    . The maximum cardinality of a quorum coloring of\n                    <jats:italic>G<\/jats:italic>\n                    is called the quorum coloring number of\n                    <jats:italic>G<\/jats:italic>\n                    and is denoted by\n                    <jats:italic>\u03c8<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>q<\/jats:italic>\n                    <\/jats:sub>\n                    (\n                    <jats:italic>G<\/jats:italic>\n                    ). A quorum coloring of order\n                    <jats:italic>\u03c8<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>q<\/jats:italic>\n                    <\/jats:sub>\n                    (\n                    <jats:italic>G<\/jats:italic>\n                    ) is a\n                    <jats:italic>\u03c8<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>q<\/jats:italic>\n                    <\/jats:sub>\n                    -coloring. In this paper, we determine the exact value of the quorum coloring number of all perfect trees and show that this value is computable in linear time. Moreover, we design a linear-time algorithm that finds a\n                    <jats:italic>\u03c8<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>q<\/jats:italic>\n                    <\/jats:sub>\n                    -coloring for any perfect tree, which partially answers an open question raised by Hedetniemi\n                    <jats:italic>et al.<\/jats:italic>\n                    [\n                    <jats:italic>AKCE Int. J. Graphs Comb.<\/jats:italic>\n                    <jats:bold>10<\/jats:bold>\n                    (2013) 97\u2013109].\n                  <\/jats:p>","DOI":"10.1051\/ro\/2025168","type":"journal-article","created":{"date-parts":[[2025,12,23]],"date-time":"2025-12-23T08:58:53Z","timestamp":1766480333000},"page":"269-280","source":"Crossref","is-referenced-by-count":0,"title":["Quorum colorings of perfect trees"],"prefix":"10.1051","volume":"60","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1175-5137","authenticated-orcid":false,"given":"Rafik","family":"Sahbi","sequence":"first","affiliation":[{"name":"Department of Fundamental Science and Technology, National Higher School of Advanced Technologies","place":["Algeria"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Asmaa","family":"Issad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Blida 1","place":["Algeria"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"250","published-online":{"date-parts":[[2026,2,25]]},"reference":[{"key":"R1","first-page":"519","volume":"103","author":"Eroh","year":"2012","journal-title":"ARS Comb."},{"key":"R2","doi-asserted-by":"crossref","first-page":"70","DOI":"10.5614\/ejgta.2014.2.1.7","volume":"2","author":"Fernaua","year":"2014","journal-title":"Electron. J. Graph Theory Appl."},{"key":"R3","first-page":"37","volume":"38","author":"Fricke","year":"2003","journal-title":"Bull. ICA"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"Haynes T.W. and Hedetniemi S.T., Alliances and related domination parameters, in Structures of Domination in Graphs. Vol. 66. Springer (2021) 47\u201377.","DOI":"10.1007\/978-3-030-58892-2_3"},{"key":"R5","first-page":"51","volume":"4","author":"Haynes","year":"2007","journal-title":"AKCE Int. J. Graphs Combin."},{"key":"R6","first-page":"157","volume":"48","author":"Hedetniemi","year":"2004","journal-title":"J. Combin. Math. Combin. Comput."},{"key":"R7","first-page":"97","volume":"10","author":"Hedetniemi","year":"2013","journal-title":"AKCE Int. J. Graphs Comb."},{"key":"R8","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.akcej.2017.05.002","volume":"15","author":"Ouazine","year":"2018","journal-title":"AKCE Int. J. Graphs Comb."},{"key":"R9","doi-asserted-by":"crossref","first-page":"784","DOI":"10.1016\/j.akcej.2019.12.010","volume":"17","author":"Sahbi","year":"2020","journal-title":"AKCE Int. J. Graphs Comb."},{"key":"R10","doi-asserted-by":"crossref","first-page":"2385","DOI":"10.1051\/ro\/2021116","volume":"55","author":"Sahbi","year":"2021","journal-title":"RAIRO Oper. Res."},{"key":"R11","doi-asserted-by":"crossref","first-page":"106297","DOI":"10.1016\/j.ipl.2022.106297","volume":"178","author":"Sahbi","year":"2022","journal-title":"Inf. Process. Lett."},{"key":"R12","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1016\/j.dam.2018.03.060","volume":"247","author":"Sahbi","year":"2018","journal-title":"Discrete Appl. Math."},{"key":"R13","unstructured":"Sahbi R., Belkina Y. and Bennadji A., Sub-quorum colorins of some infinite families of caterpillars. Submitted for publication."},{"key":"R14","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1051\/ita\/2025018","volume":"59","author":"Sahbi","year":"2025","journal-title":"RAIRO-Theor. Inf. App."},{"key":"R15","unstructured":"Shafique K.H., Partitioning a graph in alliances and its application to data clustering. Ph. D. Thesis in Computer Science, University of Central Florida (2004)."},{"key":"R16","first-page":"141","volume":"105","author":"Yero","year":"2017","journal-title":"Util. Math."},{"key":"R17","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s10114-011-9075-1","volume":"27","author":"Yero","year":"2011","journal-title":"Acta Math. Sin. (Engl. Ser.)"}],"container-title":["RAIRO - Operations Research"],"original-title":[],"link":[{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2025168\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T08:59:21Z","timestamp":1772009961000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2025168"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1]]},"references-count":17,"journal-issue":{"issue":"1"},"alternative-id":["ro250533"],"URL":"https:\/\/doi.org\/10.1051\/ro\/2025168","relation":{},"ISSN":["0399-0559","2804-7303"],"issn-type":[{"value":"0399-0559","type":"print"},{"value":"2804-7303","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1]]}}}