{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T20:19:06Z","timestamp":1772050746891,"version":"3.50.1"},"reference-count":17,"publisher":"EDP Sciences","license":[{"start":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T00:00:00Z","timestamp":1761696000000},"content-version":"vor","delay-in-days":301,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Theor. Inf. Appl."],"accepted":{"date-parts":[[2025,9,25]]},"published-print":{"date-parts":[[2025]]},"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>V<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>k<\/jats:italic>\n                    <\/jats:sub>\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>V<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>i<\/jats:italic>\n                    <\/jats:sub>\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 \u03c8\n                    <jats:sub>\n                      <jats:italic>q<\/jats:italic>\n                    <\/jats:sub>\n                    (\n                    <jats:italic>G<\/jats:italic>\n                    ). A quorum coloring of\n                    <jats:italic>G<\/jats:italic>\n                    of order \u03c8\n                    <jats:sub>\n                      <jats:italic>q<\/jats:italic>\n                    <\/jats:sub>\n                    (\n                    <jats:italic>G<\/jats:italic>\n                    ) is a \u03c8\n                    <jats:sub>\n                      <jats:italic>q<\/jats:italic>\n                    <\/jats:sub>\n                    -coloring of\n                    <jats:italic>G<\/jats:italic>\n                    . In this paper, we design a linear-time algorithm for finding both a \u03c8\n                    <jats:sub>\n                      <jats:italic>q<\/jats:italic>\n                    <\/jats:sub>\n                    -coloring and the quorum coloring number for every perfect tree whose the vertices of the same level have the same degree.\n                  <\/jats:p>","DOI":"10.1051\/ita\/2025018","type":"journal-article","created":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T08:52:29Z","timestamp":1761727949000},"page":"15","source":"Crossref","is-referenced-by-count":1,"title":["Quorum colorings of maximum cardinality in linear time for a subclass of perfect trees"],"prefix":"10.1051","volume":"59","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1175-5137","authenticated-orcid":false,"given":"Rafik","family":"Sahbi","sequence":"first","affiliation":[{"name":"Department of the Preparatory Training, National Higher School of Advanced Technologies","place":["Algeria"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wissam","family":"Boumalha","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Blida 1","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":[[2025,10,29]]},"reference":[{"key":"R1","first-page":"97","volume":"10","author":"Hedetniemi","year":"2013","journal-title":"AKCE Int. J. Graphs Comb."},{"key":"R2","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":"R3","doi-asserted-by":"crossref","unstructured":"Sahbi R., Solutions to four open problems on quorum colorings of graphs. RAIRO Oper. Res. 55 (2294) 2385\u20132394.","DOI":"10.1051\/ro\/2021116"},{"key":"R4","unstructured":"Sahbi R., Belkina Y. and Bennadji A., Sub-quorum colorins of some infinite families of caterpillars. Submitted in RAIRO: Theoretical Informatics and Applications (2023)."},{"key":"R5","unstructured":"Shafique K.H., Partitioning a graph in alliances and its application to data clustering, Ph.D. Thesis, University of Central Florida (2004)."},{"key":"R6","first-page":"519","volume":"103","author":"Eroh","year":"2012","journal-title":"Ars Combinatoria"},{"key":"R7","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":"R8","first-page":"37","volume":"38","author":"Fricke","year":"2003","journal-title":"Bull. ICA"},{"key":"R9","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/978-3-030-58892-2_3","volume":"66","author":"Haynes","year":"2021","journal-title":"Structures of Domination in Graphs"},{"key":"R10","first-page":"51","volume":"4","author":"Haynes","year":"2007","journal-title":"AKCE Int. J. Graphs Combin."},{"key":"R11","first-page":"157","volume":"48","author":"Hedetniemi","year":"2004","journal-title":"J. Combin. Math. Combin. Comput."},{"key":"R12","first-page":"157","volume":"48","author":"Hedetniemi","year":"2004","journal-title":"J. Combin. Math. Combin. Comput."},{"key":"R13","first-page":"141","volume":"105","author":"Yero","year":"2017","journal-title":"Util. Math."},{"key":"R14","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.)"},{"key":"R15","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":"R16","doi-asserted-by":"crossref","first-page":"106297","DOI":"10.1016\/j.ipl.2022.106297","volume":"178","author":"Sahbi","year":"2022","journal-title":"Inf. Process. 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