{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T17:52:53Z","timestamp":1770054773620,"version":"3.49.0"},"reference-count":29,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01","funder":[{"name":"the Postgraduate Education Reform and Quality Improvement Project of Henan Province","award":["YJS2024KC32"],"award-info":[{"award-number":["YJS2024KC32"]}]},{"name":"Young Backbone Teachers of Higher Education Institutions in Henan Province","award":["2024GGJS045"],"award-info":[{"award-number":["2024GGJS045"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Inter. Net."],"published-print":{"date-parts":[[2026,3]]},"abstract":"<jats:p>A tree [Formula: see text] in a total colored graph [Formula: see text] is called a total-monochromatic tree if all the internal vertices and edges of [Formula: see text] have the same color. For [Formula: see text], a total-monochromatic [Formula: see text]-tree in [Formula: see text] is a total-monochromatic tree of [Formula: see text] containing the vertices of [Formula: see text]. For a connected graph [Formula: see text] and a given integer [Formula: see text] with [Formula: see text], the [Formula: see text]-monochromatic total-index [Formula: see text] of [Formula: see text] is the maximum number of colors needed such that for each subset [Formula: see text] of [Formula: see text] vertices, there exists a total-monochromatic [Formula: see text]-tree. In this paper, we first show some basic results about [Formula: see text]. Next, we determine all connected graphs [Formula: see text] for which [Formula: see text] is [Formula: see text] and [Formula: see text]. Moreover, we investigate the [Formula: see text]-monochromatic total-index of a graph [Formula: see text] according to some constraints of [Formula: see text]. Finally, we determine the [Formula: see text]-monochromatic total-index of a graph [Formula: see text] with order [Formula: see text] and [Formula: see text] for [Formula: see text].<\/jats:p>","DOI":"10.1142\/s0219265924500282","type":"journal-article","created":{"date-parts":[[2024,12,17]],"date-time":"2024-12-17T20:17:48Z","timestamp":1734466668000},"source":"Crossref","is-referenced-by-count":0,"title":["The 3-Monochromatic Total-Index of a Graph"],"prefix":"10.1142","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0380-9468","authenticated-orcid":false,"given":"Yingbin","family":"Ma","sequence":"first","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Zhao","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mingli","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2024,12,16]]},"reference":[{"key":"S0219265924500282BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84628-970-5"},{"key":"S0219265924500282BIB002","doi-asserted-by":"publisher","DOI":"10.1007\/s10878-015-9938-y"},{"key":"S0219265924500282BIB003","doi-asserted-by":"publisher","DOI":"10.1007\/s10878-018-0258-x"},{"key":"S0219265924500282BIB004","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2011.04.020"},{"key":"S0219265924500282BIB005","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.20643"},{"key":"S0219265924500282BIB006","doi-asserted-by":"publisher","DOI":"10.21136\/MB.2008.133947"},{"key":"S0219265924500282BIB007","doi-asserted-by":"publisher","DOI":"10.7151\/dmgt.1780"},{"key":"S0219265924500282BIB008","doi-asserted-by":"publisher","DOI":"10.1007\/s10878-022-00850-5"},{"key":"S0219265924500282BIB009","doi-asserted-by":"publisher","DOI":"10.1007\/s40840-015-0274-2"},{"key":"S0219265924500282BIB010","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2016.08.020"},{"issue":"2","key":"S0219265924500282BIB011","first-page":"263","volume":"340","author":"Jiang H.","year":"2018","journal-title":"Ars Combin."},{"key":"S0219265924500282BIB012","doi-asserted-by":"publisher","DOI":"10.11650\/tjm\/200102"},{"key":"S0219265924500282BIB013","doi-asserted-by":"publisher","DOI":"10.7151\/dmgt.1547"},{"key":"S0219265924500282BIB014","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.20418"},{"key":"S0219265924500282BIB015","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2017.06.004"},{"key":"S0219265924500282BIB016","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-012-1204-9"},{"key":"S0219265924500282BIB017","doi-asserted-by":"publisher","DOI":"10.7151\/dmgt.1664"},{"key":"S0219265924500282BIB018","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-012-1243-2"},{"key":"S0219265924500282BIB019","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-3119-0"},{"issue":"1","key":"S0219265924500282BIB020","volume":"0","author":"Li X.","year":"2017","journal-title":"Theory Appl. Graphs"},{"key":"S0219265924500282BIB021","doi-asserted-by":"publisher","DOI":"10.1007\/s10878-016-0048-2"},{"issue":"1","key":"S0219265924500282BIB022","volume":"0","author":"Li X.","year":"2018","journal-title":"Theory Appl. Graphs"},{"key":"S0219265924500282BIB023","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2013.04.025"},{"key":"S0219265924500282BIB024","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2014.04.012"},{"key":"S0219265924500282BIB025","doi-asserted-by":"publisher","DOI":"10.1007\/s00025-015-0469-8"},{"key":"S0219265924500282BIB026","doi-asserted-by":"publisher","DOI":"10.1007\/s40840-020-00992-x"},{"key":"S0219265924500282BIB027","first-page":"127128","volume":"427","author":"Ma Y.","year":"2022","journal-title":"Appl. Math. Comput."},{"key":"S0219265924500282BIB028","doi-asserted-by":"publisher","DOI":"10.7151\/dmgt.1553"},{"key":"S0219265924500282BIB029","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2011.05.001"}],"container-title":["Journal of Interconnection Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0219265924500282","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T07:00:10Z","timestamp":1770015610000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S0219265924500282"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,16]]},"references-count":29,"journal-issue":{"issue":"01","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["10.1142\/S0219265924500282"],"URL":"https:\/\/doi.org\/10.1142\/s0219265924500282","relation":{},"ISSN":["0219-2659","1793-6713"],"issn-type":[{"value":"0219-2659","type":"print"},{"value":"1793-6713","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,16]]},"article-number":"2450028"}}